Compiled and Solved Problems in Geometry and Trigonometry

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Florentin Smarandache

Compiled and Solved Problems in Geometry and Trigonometry

255 Compiled and Solved Problems in Geometry and Trigonometry

FLORENTIN SMARANDACHE

255 Compiled and Solved Problems in Geometry and Trigonometry (from Romanian Textbooks)

Educational Publisher 2015

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Florentin Smarandache

Peer reviewers: Prof. Rajesh Singh, School of Statistics, DAVV, Indore (M.P.), India. Dr. Linfan Mao, Academy of Mathematics and Systems, Chinese Academy of Sciences, Beijing 100190, P. R. China. Mumtaz Ali, Department of Mathematics, Quaid-i-Azam University, Islamabad, 44000, Pakistan Prof. Stefan Vladutescu, University of Craiova, Romania. Said Broumi, University of Hassan II Mohammedia, Hay El Baraka Ben M'sik, Casablanca B. P. 7951, Morocco.

E-publishing, Translation & Editing: Dana Petras, Nikos Vasiliou AdSumus Scientific and Cultural Society, Cantemir 13, Oradea, Romania

Copyright: Florentin Smarandache 1998-2015 Educational Publisher, Chicago, USA ISBN: 978-1-59973-299-2

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255 Compiled and Solved Problems in Geometry and Trigonometry

Table of Content Explanatory Note ..................................................................................................................................................... 4 Problems in Geometry (9th grade) ................................................................................................................... 5 Solutions ............................................................................................................................................................... 11 Problems in Geometry and Trigonometry ................................................................................................. 38 Solutions ............................................................................................................................................................... 42 Other Problems in Geometry and Trigonometry (10th grade) .......................................................... 60 Solutions ............................................................................................................................................................... 67 Various Problems................................................................................................................................................... 96 Solutions ............................................................................................................................................................... 99 Problems in Spatial Geometry ...................................................................................................................... 108 Solutions ............................................................................................................................................................ 114 Lines and Planes ................................................................................................................................................. 140 Solutions ............................................................................................................................................................ 143 Projections ............................................................................................................................................................. 155 Solutions ............................................................................................................................................................ 159 Review Problems................................................................................................................................................. 174 Solutions ............................................................................................................................................................ 182

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Florentin Smarandache

Explanatory Note

This book is a translation from Romanian of "Probleme Compilate şi Rezolvate de Geometrie şi Trigonometrie" (University of Kishinev Press, Kishinev, 169 p., 1998), and includes problems of 2D and 3D Euclidean geometry plus trigonometry, compiled and solved from the Romanian Textbooks for 9th and 10th grade students, in the period 1981-1988, when I was a professor of mathematics at the "Petrache Poenaru" National College in Balcesti, Valcea (Romania), Lycée Sidi El Hassan Lyoussi in Sefrou (Morroco), then at the "Nicolae Balcescu" National College in Craiova and Dragotesti General School (Romania), but also I did intensive private tutoring for students preparing their university entrance examination. After that, I have escaped in Turkey in September 1988 and lived in a political refugee camp in Istanbul and Ankara, and in March 1990 I immigrated to United States. The degree of difficulties of the problems is from easy and medium to hard. The solutions of the problems are at the end of each chapter. One can navigate back and forth from the text of the problem to its solution using bookmarks. The book is especially a didactical material for the mathematical students and instructors. The Author

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255 Compiled and Solved Problems in Geometry and Trigonometry

Problems in Geometry (9

th

1.

grade)

The measure of a regular polygon’s interior angle is four times bigger than the measure of its external angle. How many sides does the polygon have? Solution to Problem 1

2. How many sides does a convex polygon have if all its external angles are obtuse? Solution to Problem 2

3. Show that in a convex quadrilateral the bisector of two consecutive angles forms an angle whose measure is equal to half the sum of the measures of the other two angles. Solution to Problem 3

4. Show that the surface of a convex pentagon can be decomposed into two quadrilateral surfaces. Solution to Problem 4

5. What is the minimum number of quadrilateral surfaces in which a convex polygon with 9, 10, 11 vertices can be decomposed? Solution to Problem 5

′ đ??ľ ′ đ??ś ′ ), then ∃ bijective function đ?‘“ = (đ??´đ??ľđ??ś) ′ đ??ľ ′ đ??ś ′ ) such Ě‚ ≥ (đ??´Ě‚ Ě‚ → (đ??´Ě‚ 6. If (đ??´đ??ľđ??ś)

Ě‚ , ‖đ?‘ƒđ?‘„‖ = ‖đ?‘“(đ?‘ƒ)‖, ‖đ?‘“(đ?‘„)‖, and vice versa. that for ∀ 2 points đ?‘ƒ, đ?‘„ ∈ (đ??´đ??ľđ??ś) Solution to Problem 6

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7. If ∆đ??´đ??ľđ??ś ≥ ∆đ??´â€˛ đ??ľ ′ đ??ś ′ then ∃ bijective function đ?‘“ = đ??´đ??ľđ??ś → đ??´â€˛ đ??ľ ′ đ??ś ′ such that (∀) 2 points đ?‘ƒ, đ?‘„ ∈ đ??´đ??ľđ??ś, ‖đ?‘ƒđ?‘„‖ = ‖đ?‘“(đ?‘ƒ)‖, ‖đ?‘“(đ?‘„)‖, and vice versa. Solution to Problem 7

8. Show that if ∆đ??´đ??ľđ??ś~∆đ??´â€˛ đ??ľ ′ đ??ś ′ , then [đ??´đ??ľđ??ś]~[đ??´â€˛ đ??ľ ′ đ??ś ′ ]. Solution to Problem 8

9. Show that any two rays are congruent sets. The same property for lines. Solution to Problem 9

10. Show that two disks with the same radius are congruent sets. Solution to Problem 10

11. If the function đ?‘“: đ?‘€ → đ?‘€â€˛ is isometric, then the inverse function đ?‘“ −1 : đ?‘€ → đ?‘€â€˛ is as well isometric. Solution to Problem 11

12. If the convex polygons đ??ż = đ?‘ƒ1 , đ?‘ƒ2 , ‌ , đ?‘ƒđ?‘› and đ??żâ€˛ = đ?‘ƒ1′ , đ?‘ƒ2′ , ‌ , đ?‘ƒđ?‘›â€˛ have |đ?‘ƒđ?‘– , đ?‘ƒđ?‘–+1 | ≥ ′ ′̂′ Ě‚ |đ?‘ƒđ?‘–′ , đ?‘ƒđ?‘–+1 | for đ?‘– = 1, 2, ‌ , đ?‘› − 1, and đ?‘ƒđ?‘– đ?‘ƒđ?‘–+1 đ?‘ƒđ?‘–+2 ≥ đ?‘ƒđ?‘–′ đ?‘ƒđ?‘–+1 đ?‘ƒđ?‘–+2 , (∀) đ?‘– = 1, 2, ‌ , đ?‘› −

2, then đ??ż ≥ đ??żâ€˛ and [đ??ż] ≥ [đ??żâ€˛ ]. Solution to Problem 12

13. Prove that the ratio of the perimeters of two similar polygons is equal to their similarity ratio. Solution to Problem 13

14. The parallelogram đ??´đ??ľđ??śđ??ˇ has ‖đ??´đ??ľâ€– = 6, ‖đ??´đ??śâ€– = 7 and đ?‘‘(đ??´đ??ś) = 2. Find đ?‘‘(đ??ˇ, đ??´đ??ľ). Solution to Problem 14

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255 Compiled and Solved Problems in Geometry and Trigonometry

15. Of triangles đ??´đ??ľđ??ś with ‖đ??ľđ??śâ€– = đ?‘Ž and ‖đ??śđ??´â€– = đ?‘?, đ?‘Ž and đ?‘? being given numbers, find a triangle with maximum area. Solution to Problem 15

16. Consider a square đ??´đ??ľđ??śđ??ˇ and points đ??¸, đ??š, đ??ş, đ??ť, đ??ź, đ??ž, đ??ż, đ?‘€ that divide each side in three congruent segments. Show that đ?‘ƒđ?‘„đ?‘…đ?‘† is a square and its area is 2

equal to 9 đ?œŽ[đ??´đ??ľđ??śđ??ˇ]. Solution to Problem 16

17. The diagonals of the trapezoid đ??´đ??ľđ??śđ??ˇ (đ??´đ??ľ||đ??ˇđ??ś) cut at đ?‘‚. a. Show that the triangles đ??´đ?‘‚đ??ˇ and đ??ľđ?‘‚đ??ś have the same area; b. The parallel through đ?‘‚ to đ??´đ??ľ cuts đ??´đ??ˇ and đ??ľđ??ś in đ?‘€ and đ?‘ . Show that ||đ?‘€đ?‘‚|| = ||đ?‘‚đ?‘ ||. Solution to Problem 17

18. đ??¸ being the midpoint of the non-parallel side [đ??´đ??ˇ] of the trapezoid đ??´đ??ľđ??śđ??ˇ, show that đ?œŽ[đ??´đ??ľđ??śđ??ˇ] = 2đ?œŽ[đ??ľđ??śđ??¸]. Solution to Problem 18

Ě‚ and a point đ??ˇ inside the angle. A line 19. There are given an angle (đ??ľđ??´đ??ś) through đ??ˇ cuts the sides of the angle in đ?‘€ and đ?‘ . Determine the line đ?‘€đ?‘ such that the area ∆đ??´đ?‘€đ?‘ to be minimal. Solution to Problem 19

20. Construct a point đ?‘ƒ inside the triangle đ??´đ??ľđ??ś, such that the triangles đ?‘ƒđ??´đ??ľ, đ?‘ƒđ??ľđ??ś, đ?‘ƒđ??śđ??´ have equal areas. Solution to Problem 20

21. Decompose a triangular surface in three surfaces with the same area by parallels to one side of the triangle. Solution to Problem 21 7

Florentin Smarandache

22. Solve the analogous problem for a trapezoid. Solution to Problem 22

23. We extend the radii drawn to the peaks of an equilateral triangle inscribed in a circle đ??ż(đ?‘‚, đ?‘&#x;), until the intersection with the circle passing through the peaks of a square circumscribed to the circle đ??ż(đ?‘‚, đ?‘&#x;). Show that the points thus obtained are the peaks of a triangle with the same area as the hexagon inscribed in đ??ż(đ?‘‚, đ?‘&#x;). Solution to Problem 23

24. Prove the leg theorem with the help of areas. Solution to Problem 24

25. Consider an equilateral ∆đ??´đ??ľđ??ś with ‖đ??´đ??ľâ€– = 2đ?‘Ž. The area of the shaded surface determined by circles đ??ż(đ??´, đ?‘Ž), đ??ż(đ??ľ, đ?‘Ž), đ??ż(đ??´, 3đ?‘Ž) is equal to the area of Ě‚ of the circle đ??ż(đ??ś, đ?‘Ž). the circle sector determined by the minor arc (đ??¸đ??š) Solution to Problem 25

26. Show that the area of the annulus between circles đ??ż(đ?‘‚, đ?‘&#x;2 ) and đ??ż(đ?‘‚, đ?‘&#x;2 ) is equal to the area of a disk having as diameter the tangent segment to circle đ??ż(đ?‘‚, đ?‘&#x;1 ) with endpoints on the circle đ??ż(đ?‘‚, đ?‘&#x;2 ). Solution to Problem 26

27. Let [đ?‘‚đ??´], [đ?‘‚đ??ľ] two ⊼ radii of a circle centered at [đ?‘‚]. Take the points đ??ś and Ě‚ such that đ??´đ??ś Ě‚ ≥đ??ľđ??ˇ Ě‚ and let đ??¸, đ??š be the projections of đ??ˇ on the minor arc đ??´đ??ľđ??š đ??śđ??ˇ onto đ?‘‚đ??ľ. Show that the area of the surface bounded by [đ??ˇđ??š], [đ??šđ??¸[đ??¸đ??ś]] Ě‚ is equal to the area of the sector determined by arc đ??śđ??ˇ Ě‚ of the and arc đ??śđ??ˇ circle đ??ś(đ?‘‚, ‖đ?‘‚đ??´â€–). Solution to Problem 27

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255 Compiled and Solved Problems in Geometry and Trigonometry

28. Find the area of the regular octagon inscribed in a circle of radius đ?‘&#x;. Solution to Problem 28

29. Using areas, show that the sum of the distances of a variable point inside the equilateral triangle đ??´đ??ľđ??ś to its sides is constant. Solution to Problem 29

30. Consider a given triangle đ??´đ??ľđ??ś and a variable point đ?‘€ ∈ |đ??ľđ??ś|. Prove that between the distances đ?‘Ľ = đ?‘‘(đ?‘€, đ??´đ??ľ) and đ?‘Ś = đ?‘‘(đ?‘€, đ??´đ??ś) is a relation of đ?‘˜đ?‘Ľ + đ?‘™đ?‘Ś = 1 type, where đ?‘˜ and đ?‘™ are constant. Solution to Problem 30

31. Let đ?‘€ and đ?‘ be the midpoints of sides [đ??ľđ??ś] and [đ??´đ??ˇ] of the convex quadrilateral đ??´đ??ľđ??śđ??ˇ and {đ?‘ƒ} = đ??´đ?‘€ ∊ đ??ľđ?‘ and {đ?‘„} = đ??śđ?‘ ∊ đ?‘ đ??ˇ. Prove that the area of the quadrilateral đ?‘ƒđ?‘€đ?‘„đ?‘ is equal to the sum of the areas of triangles đ??´đ??ľđ?‘ƒ and đ??śđ??ˇđ?‘„. Solution to Problem 31

32. Construct a triangle having the same area as a given pentagon. Solution to Problem 32

33. Construct a line that divides a convex quadrilateral surface in two parts with equal areas. Solution to Problem 33

34. In a square of side đ?‘™, the middle of each side is connected with the ends of the opposite side. Find the area of the interior convex octagon formed in this way. Solution to Problem 34

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Florentin Smarandache

35. The diagonal [đ??ľđ??ˇ] of parallelogram đ??´đ??ľđ??śđ??ˇ is divided by points đ?‘€, đ?‘ , in 3 segments. Prove that đ??´đ?‘€đ??śđ?‘ is a parallelogram and find the ratio between đ?œŽ[đ??´đ?‘€đ??śđ?‘ ] and đ?œŽ[đ??´đ??ľđ??śđ??ˇ]. Solution to Problem 35

36. There are given the points đ??´, đ??ľ, đ??ś, đ??ˇ, such that đ??´đ??ľ ∊ đ??śđ??ˇ = {đ?‘?}. Find the locus of point đ?‘€ such that đ?œŽ[đ??´đ??ľđ?‘€] = đ?œŽ[đ??śđ??ˇđ?‘€]. Solution to Problem 36

37. Analogous problem for đ??´đ??ľ||đ??śđ??ˇ. Solution to Problem 37

38. Let đ??´đ??ľđ??śđ??ˇ be a convex quadrilateral. Find the locus of point đ?‘Ľ1 inside đ??´đ??ľđ??śđ??ˇ such that đ?œŽ[đ??´đ??ľđ?‘€] + đ?œŽ[đ??śđ??ˇđ?‘€] = đ?‘˜, đ?‘˜ – a constant. For which values of đ?‘˜ the desired geometrical locus is not the empty set? Solution to Problem 38

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255 Compiled and Solved Problems in Geometry and Trigonometry

Solutions Solution to Problem 1.

180 (đ?‘› − 2) 180 =4 ďƒž đ?‘› = 10 đ?‘› 5

Solution to Problem 2. đ?‘Ľ1 > 900 Let đ?‘› = 3 â&#x;š đ?‘Ľ2 > 900 } â&#x;š đ?‘Ľ1 + đ?‘Ľ2 + đ?‘Ľ3 > 2700, so đ?‘› = 3 is possible. đ?‘Ľ1 , đ?‘Ľ2 , đ?‘Ľ3 âˆ˘ ext đ?‘Ľ3 > 900 Let đ?‘› = 4 â&#x;š đ?‘Ľ1 , đ?‘Ľ2 , đ?‘Ľ3 , đ?‘Ľ4 âˆ˘ ext

đ?‘Ľ1 > 900 } â&#x;š đ?‘Ľ1 + đ?‘Ľ2 + đ?‘Ľ3 + đ?‘Ľ4 > 3600 , so đ?‘› = 4 is impossible. â‹Ž 0 đ?‘Ľ3 > 90

Therefore, đ?‘› = 3.

Solution to Problem 3.

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Florentin Smarandache

Ě‚ ) + m(đ??śĚ‚ ) m(đ??ˇ 2 Ě‚ ) = 360° m(đ??´Ě‚) + m(đ??ľĚ‚) + m(đ??śĚ‚ ) + m(đ??ˇ Ě‚) m(đ??´Ě‚) + m(đ??ľĚ‚) m(đ??śĚ‚ ) + m(đ??ˇ = 180° − 2 2 m(đ??´Ě‚) m(đ??ľĚ‚) Ě‚ ) = 180° − m(đ??´đ??¸đ??ľ − = 2 2 Ě‚) Ě‚) m(đ??śĚ‚ ) + m(đ??ˇ m(đ??śĚ‚ ) + (đ??ˇ = 180° − 180° + = 2 2 Ě‚) = m(đ??´đ??¸đ??ľ

Solution to Problem 4. Ě‚ ⇒ đ??´, đ??ľ ∈ int. đ??¸đ??ˇđ??ś Ě‚ . Let đ?‘€ ∈ |đ??´đ??ľ| ⇒ đ?‘€ ∈ int. đ??¸đ??ˇđ??ś Ě‚ ⇒ |đ??ˇđ?‘€ ⊂ int. đ??¸đ??ˇđ??ś Ě‚ , |đ??¸đ??´| ∊ Let đ??¸đ??ˇđ??ś |đ??ˇđ?‘€ = ∅ ⇒ đ??ˇđ??¸đ??´đ?‘€ quadrilateral. The same for đ??ˇđ??śđ??ľđ?‘€.

Solution to Problem 5.

9 vertices; 4 quadrilaterals.

10 vertices; 4 quadrilaterals.

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11 vertices; 5 quadrilaterals.

255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 6. Ě‚ such that Ě‚ . We construct a function đ?‘“:Ě‚ Ě‚ ≥ A’B’C’ We assume that ABC đ??´đ??ľđ??ś → đ??´â€˛đ??ľâ€˛đ??śâ€˛ ′ đ?‘“ (B) = B { if P ∈ |BA, đ?‘“(P) ∈ B′A′ đ?‘ƒ ∈ |đ??ľđ??ś, đ?‘“(đ?‘ƒ) ∈ đ??ľâ€˛ đ??ś ′ such that ‖đ??ľđ?‘ƒâ€– = ‖đ??ľâ€˛ đ?‘ƒâ€˛ ‖ where đ?‘ƒâ€˛ = đ?‘“(đ??š). The so constructed function is bijective, since for different arguments there are different corresponding values and ∀ point from đ??´â€˛đ??ľâ€˛đ??śâ€˛ is the image of a single Ě‚ (from the axiom of segment construction). point from đ??´đ??ľđ??ś

If đ?‘ƒ, đ?‘„ ∈ this ray, ‖BP‖ = ‖B ′ P′ ‖ } â&#x;š ‖PQ‖ = ‖BQ‖ − ‖BP‖ = ‖B′Q′‖ − ‖B ′ P′ ‖ = ‖P′Q′‖ = ‖đ?‘“(P), đ?‘“(Q)‖. ‖BQ‖ = ‖B′Q′‖ If đ?‘ƒ, đ?‘„ ∈ a different ray, ‖BP‖ = ‖B ′ P′ ‖ ‖BQ‖ = ‖B′Q′‖} â&#x;š ∆PBQ = ∆P′ B ′ Q′ ďƒžâ€–PQ‖ = ‖P′ Q′ ‖ = ‖đ?‘“ (P), đ?‘“ (Q)‖. ′ B′Q′ Ě‚ ≥ PĚ‚ PBQ

Vice versa. Let đ?‘“ âˆś ABC → A′ B ′ C′ such that đ?‘“ bijective and ‖PQ‖ = ‖đ?‘“(P), đ?‘“(Q)‖.

Let đ?‘ƒ, đ?‘„ ∈ |đ??ľđ??´ and đ?‘…đ?‘† ∈ |đ??ľđ??ś. 13

Florentin Smarandache

‖PQ‖ = ‖P′ Q′ ‖ ′ P ′ S ′ ⟹ BPS ′ P′ S′ ̂ ≡ B̂ ‖PS‖ = ‖P′ S ′ ‖ } ⟹ ∆PQS ≡ ∆P′ Q′ S ′ ⟹ ̂ QPS ≡ Q̂ ‖QS‖ = ‖Q′S′‖ ‖𝑃𝑆‖ = ‖𝑃′𝑆′‖ ̂ (2). ̂ ≡ P′S′B′ ‖𝑅𝑃‖ = ‖𝑅′𝑃′‖} ⟹ ∆PRS ≡ ∆P′R′S′′ ⟹ PSB ‖𝑃𝑆‖ = ‖𝑃′𝑆′‖ ̂ (as diff. at 180°) i.e. ABC ̂. ̂ ≡ P′B′S′ ̂ ≡ A′B′C′ From (1) and (2) ⟹ PBC

(1);

Solution to Problem 7.

Let ∆𝐴𝐵𝐶 ≡ ∆𝐴′𝐵′𝐶′. We construct a function 𝑓 ∶ 𝐴𝐵𝐶 → 𝐴′𝐵′𝐶′ such that 𝑓(𝐴) = 𝐴′, 𝑓(𝐵) = 𝐵′, 𝑓(𝐶) = 𝐶′ and so

𝑃 ∈ |𝐴𝐵| → 𝑃′ = 𝑓(𝑃) ∈ |𝐴′𝐵′| such that ||𝐴𝑃|| = ||𝐴′𝑃′||; 𝑃 ∈ |𝐵𝐶| → 𝑃′ = 𝑓(𝑃) ∈ |𝐵′𝐶′| such that ||𝐵𝑃|| = ||𝐵′𝑃′||; 𝑃 ∈ |𝐶𝐴| → 𝑃′ = 𝑓(𝑃) ∈ |𝐶′𝐴′| such that ||𝐶𝑃|| = ||𝐶′𝑃′||.

The so constructed function is bijective. Let 𝑃 ∈ |𝐴𝐵| and 𝑎 ∈ |𝐶𝐴| ⟹ 𝑃′ ∈ |𝐴′𝐵′| and 𝑄′ ∈ |𝐶′𝐴′|. ‖AP‖ = ‖A′P′‖ ‖CQ‖ = ‖C′Q′‖} ⟹ ‖AQ‖ = ‖A′Q′‖; A ≡ A′ ⟹ ∆APQ ≡ ∆A′P′Q′ ⟹ ‖PQ‖ = ‖P′Q′‖. ‖CA‖ = ‖C′A′‖ Similar reasoning for (∀) point 𝑃 and 𝑄.

Vice versa. We assume that ∃ a bijective function 𝑓 ∶ 𝐴𝐵𝐶 → 𝐴′𝐵′𝐶′ with the stated properties. We denote 𝑓(A) = A′′ , 𝑓(B) = B ′′ , 𝑓(C) = C ′′ ⟹ ‖AB‖ = ‖A′′ B ′′ ‖, ‖BC‖ = ‖B ′′ C′′ ‖, ‖AC‖ = ‖A′′ C′′ ‖∆ABC = ∆A′′ B ′′ C ′′ . Because 𝑓(ABC) = 𝑓([AB] ∪ [BC] ∪ [CA]) = 𝑓([AB]) ∪ 𝑓([BC]) ∪ 𝑓([CA]) = [A′′B′′] ∪ [B ′′ C ′′ ][C′′ A′′ ] = A′′ B ′′ C ′′ . But by the hypothesis 𝑓(𝐴𝐵𝐶) = 𝑓(𝐴’𝐵’𝐶’), therefore A′′ B ′′ C ′′ = ∆A′ B ′ C ′ ⟹ ∆ABC ≡ ∆A′ B ′ C ′ . 14

255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 8. If ∆đ??´đ??ľđ??ś~∆đ??´â€™đ??ľâ€™đ??śâ€™ then (∀) đ?‘“: đ??´đ??ľđ??ś → đ??´â€™đ??ľâ€™đ??śâ€™ and đ?‘˜ > 0 such that: ||đ?‘ƒđ?‘„|| = đ?‘˜ ||đ?‘“(đ?‘ƒ), đ?‘“(đ?‘„)||, đ?‘ƒ, đ?‘„ ∈ đ??´đ??ľđ??ś; ∆đ??´đ??ľđ??ś~∆đ??´â€˛đ??ľâ€˛đ??śâ€˛ â&#x;š

‖AB‖ ‖A′B′‖

‖BC‖

‖CA‖

= ‖B′C′‖ = ‖C′A′‖ = đ?‘˜

̂ ̂ ≥ Â′ ; B ̂ ≥ B̂′ ; Ĉ ≥ C′ A

‖AB‖ = đ?‘˜â€–A′B′‖ } â&#x;š ‖BC‖ = đ?‘˜â€–B′C′‖ . ‖CA‖ = đ?‘˜â€–C′A′‖

We construct a function đ?‘“: đ??´đ??ľđ??ś → đ??´â€˛đ??ľâ€˛đ??śâ€˛ such that đ?‘“(đ??´) = đ??´â€˛ , đ?‘“(đ??ľ) = đ??ľâ€˛ , đ?‘“(đ??ś) = đ??śâ€˛; if đ?‘ƒ ∈ |đ??ľđ??ś| → đ?‘ƒ ∈ |đ??ľâ€˛đ??śâ€˛| such that ||đ??ľđ?‘ƒ|| = đ?‘˜||đ??ľâ€˛đ?‘ƒâ€˛||; if đ?‘ƒ ∈ |đ??śđ??´| → đ?‘ƒ ∈ |đ??śâ€˛đ??´â€˛| such that ||đ??śđ?‘ƒ|| = đ?‘˜||đ??śâ€˛đ?‘ƒâ€˛||; đ?‘˜ – similarity constant.

Let đ?‘ƒ, đ?‘„ ∈ đ??´đ??ľ such that đ?‘ƒ ∈ |đ??ľđ??ś|, đ?‘„ ∈ |đ??´đ??ś| â&#x;š đ?‘ƒâ€˛ ∈ |đ??ľâ€˛đ??śâ€˛| and ||đ??ľđ?‘ƒ|| = đ?‘˜||đ??ľâ€˛đ?‘ƒâ€˛|| Q′ ∈ |A′ C ′ | and ‖CQ‖ = đ?‘˜â€–C′Q′‖ (1); As ‖BC‖ = đ?‘˜â€–B ′ C ′ ‖ â&#x;š ‖PC‖ = ‖BC‖ − ‖BP‖ = đ?‘˜â€–B ′ C ′ ‖ − đ?‘˜â€–B ′ P′ ‖ = = đ?‘˜(‖B′C′‖ − ‖B′P′‖) = đ?‘˜â€–P′C′‖ (2); CĚ‚ ≥ Ĉ′ (3). From (1), (2), and (3) â&#x;š ∆PCQ~∆P′ C′ Q′ â&#x;š ‖PQ‖ = đ?‘˜â€–P′Q′‖ . Similar reasoning for đ?‘ƒ, đ?‘„ ∈ đ??´đ??ľđ??ś. We also extend the bijective function previously constructed to the interiors of the two triangles in the following way:

Let đ?‘ƒ ∈ int. đ??´đ??ľđ??ś and we construct đ?‘ƒâ€™ ∈ int. đ??´â€™đ??ľâ€™đ??śâ€™ such that ||đ??´đ?‘ƒ|| = đ?‘˜||đ??´â€™đ?‘ƒâ€™|| (1). Ě‚ and ||đ??´đ?‘„|| = đ?‘˜||đ??´â€™đ?‘„’|| (2). Ě‚ ≥ B’A’Q’ Let đ?‘„ ∈ int. đ??´đ??ľđ??ś → đ?‘„′ ∈ int. đ??´â€™đ??ľâ€™đ??śâ€™ such that BAQ 15

Florentin Smarandache

From (1) and (2), AP AQ ′ A′ Q′ â&#x;š ∆APQ ~ ∆A′ P ′ Q′ â&#x;š ‖PQ‖ = đ?‘˜â€–P ′ Q′ ‖, Ě‚ ≥ PĚ‚ = = đ?‘˜, PAQ A′P′ A′Q′ but đ?‘ƒ, đ?‘„ ∈ [đ??´đ??ľđ??ś], so [đ??´đ??ľđ??ś] ~[đ??´â€˛đ??ľâ€˛đ??śâ€˛].

Solution to Problem 9. a. Let |đ?‘‚đ??´ and |đ?‘‚′đ??´â€˛ be two rays:

Let đ?‘“: |đ?‘‚đ??´ → |đ?‘‚′đ??´â€˛ such that đ?‘“(đ?‘‚) = đ?‘‚′ and đ?‘“(đ?‘ƒ) = đ?‘ƒâ€˛ with ||đ?‘‚đ?‘ƒ|| = ||đ?‘‚′đ?‘ƒâ€˛|. The so constructed point đ?‘ƒâ€˛ is unique and so if đ?‘ƒ ≠đ?‘„ â&#x;š ||đ?‘‚đ?‘ƒ|| ≠||đ?‘‚đ?‘„|| â&#x;š ||đ?‘‚′đ?‘ƒâ€˛|| ≠||đ?‘‚′đ?‘„′|| â&#x;š đ?‘ƒâ€˛ ≠đ?‘„′ and (∀)đ?‘ƒâ€˛ ∈ |đ?‘‚′đ??´â€˛ (∃) a single point đ?‘ƒ ∈ |đ?‘‚đ??´ such that ||đ?‘‚đ?‘ƒ|| = ||đ?‘‚′đ?‘ƒâ€˛||. The constructed function is bijective. If đ?‘ƒ, đ?‘„ ∈ |đ?‘‚đ??´, đ?‘ƒ ∈ |đ?‘‚đ?‘„| → đ?‘ƒâ€˛ đ?‘„ ′ ∈ |đ?‘‚′ đ??´â€˛ such that ‖OP‖ = ‖O′P′‖; ‖OQ‖ = ‖O′ Q′ ‖ â&#x;š ‖PQ‖ = ‖OQ‖ − ‖OP‖ = ‖O′ Q′ ‖ − ‖O′ P ′ ‖ = ‖P′ Q′ ‖(∀)P; đ?‘„ ∈ |OA â&#x;š the two rays are congruent.

b. Let � and �′ be two lines.

Let đ?‘‚ ∈ đ?‘‘ and đ?‘‚′ ∈ đ?‘‘′. We construct a function đ?‘“: đ?‘‘ → đ?‘‘′ such that đ?‘“(đ?‘‚) = đ?‘‚′ and đ?‘“ (|đ?‘‚đ??´) = |đ?‘‚′đ??´â€˛ and đ?‘“ (|đ?‘‚đ??ľ) = |đ?‘‚′đ??ľâ€˛ as at the previous point. It is proved in the same way that đ?‘“ is bijective and that ||đ?‘ƒđ?‘„|| = ||đ?‘ƒâ€˛đ?‘„′|| when đ?‘ƒ and đ?‘„ belong to the same ray.

16

255 Compiled and Solved Problems in Geometry and Trigonometry

If đ?‘ƒ, đ?‘„ belong to different rays: ‖đ?‘‚đ?‘ƒâ€– = ‖đ?‘‚′đ?‘ƒâ€˛â€– } â&#x;š ‖đ?‘ƒđ?‘„‖ = ‖đ?‘‚đ?‘ƒâ€– + ‖đ?‘‚đ?‘„‖ = ‖đ?‘‚′ đ?‘ƒâ€˛ ‖ + ‖đ?‘‚′ đ?‘„ ′ ‖ = ‖đ?‘ƒâ€˛đ?‘„′‖ ‖đ?‘‚đ?‘„‖ = ‖đ?‘‚′đ?‘„′‖

and so the two rays are congruent.

Solution to Problem 10.

We construct a function đ?‘“: đ??ˇ → đ??ˇâ€˛ such that đ?‘“(đ?‘‚) = đ?‘‚′, đ?‘“(đ??´) = đ??´â€˛ and a point (∀) đ?‘ƒ ∈ đ??ˇ → đ?‘ƒâ€˛ ∈ đ??ˇâ€˛ which are considered to be positive. From the axiom of segment and angle construction â&#x;š that the so constructed function is bijective, establishing a biunivocal correspondence between the elements of the two sets. ′ đ?‘‚′ đ?‘„ ′ . Ě‚ ≥ đ??´Ě‚ Let đ?‘„ ∈ đ??ˇ → đ?‘„′ ∈ đ??ˇâ€˛ such that ||đ?‘‚đ?‘„′|| = ||đ?‘‚đ?‘„||; đ??´đ?‘‚đ?‘„

As: ‖đ?‘‚đ?‘ƒâ€– = ‖đ?‘‚′đ?‘ƒâ€˛â€– ′ đ?‘‚′ đ?‘„ ′ â&#x;š ‖đ?‘ƒđ?‘„‖ = ‖đ?‘ƒâ€˛ đ?‘„ ′ ‖, (∀) đ?‘ƒ, đ?‘„ ∈ đ??ˇ â&#x;š đ??ˇ ≥ đ??ˇ ′ . ‖đ?‘‚đ?‘„‖ = ‖đ?‘‚′đ?‘„′‖} â&#x;š ∆đ?‘‚đ?‘ƒđ?‘„ ≥ đ?‘ƒĚ‚ ′ đ?‘‚′ đ?‘„ ′ Ě‚ ≥ đ?‘ƒĚ‚ đ?‘ƒđ?‘‚đ?‘„

Solution to Problem 11.

17

Florentin Smarandache

đ?‘“: đ?‘€ → đ?‘€â€˛ is an isometry â&#x;š đ?‘“ is bijective and (∀) đ?‘ƒ, đ?‘„ ∈ đ?‘€ we have ||đ?‘ƒđ?‘„|| = ||đ?‘“(đ?‘ƒ), đ?‘“(đ?‘„)||, đ?‘“ – bijective â&#x;š đ?‘“ – invertible and đ?‘“ −1 – bijective. ‖đ?‘ƒâ€˛đ?‘„′‖ = ‖đ?‘“(đ?‘ƒ); đ?‘“(đ?‘„)‖ = ‖đ?‘ƒđ?‘„‖ }â&#x;š đ?‘“ = ‖đ?‘“ −1 (đ?‘“(đ?‘ƒ)), đ?‘“ −1 (đ?‘“(đ?‘„))‖ = ‖đ?‘ƒđ?‘„‖ ‖đ?‘ƒâ€˛đ?‘„′‖ = ‖đ?‘“ −1 (đ?‘“(đ?‘ƒâ€˛)), đ?‘“ −1 (đ?‘“(đ?‘„′))‖, (∀)đ?‘ƒâ€˛ , đ?‘„ ′ ∈ đ?‘€,

‖đ?‘“ −1 (đ?‘ƒâ€˛);

−1 (đ?‘„′)‖

therefore đ?‘“ −1 : đ?‘€â€˛ → đ?‘€ is an isometry.

Solution to Problem 12. We construct a function đ?‘“ such that đ?‘“(đ?‘ƒđ?‘– ) = đ?‘ƒđ?‘–′ , đ?‘– = 1, 2, ‌ , đ?‘›, and if đ?‘ƒ ∈ |đ?‘ƒđ?‘– , đ?‘ƒđ?‘–+1 |.

The previously constructed function is also extended inside the polygon as Ě‚ follows: Let đ?‘‚ ∈ int. đ??ż → đ?‘‚′ ∈ int. đ??żâ€˛ such that đ?‘‚đ?‘ƒĚ‚ đ?‘– đ?‘ƒđ?‘–+1 ≥ đ?‘‚′đ?‘ƒâ€˛đ?‘– đ?‘ƒâ€˛đ?‘–+1 and ‖đ?‘‚đ?‘ƒđ?‘– ‖ = ‖đ?‘‚′đ?‘ƒâ€˛đ?‘– ‖. We connect these points with the vertices of the polygon. It can be easily proved that the triangles thus obtained are congruent. We construct the function đ?‘”: [đ??ż] → [đ??żâ€˛] such that đ?‘“(đ?‘ƒ), if đ?‘ƒ ∈ đ??ż đ?‘”(đ?‘ƒ) = { đ?‘‚′ , if đ?‘ƒ = đ?‘‚ đ?‘ƒâ€˛ , if đ?‘ƒ ∈ [đ?‘ƒđ?‘– đ?‘‚đ?‘ƒđ?‘–+1 ] such that ′ ′ ′ Ě‚ Ě‚ đ?‘ƒ đ?‘– đ?‘‚đ?‘ƒ ≥ đ?‘ƒđ?‘– đ?‘‚ đ?‘ƒ (∀)đ?‘– = 1, 2, ‌ , đ?‘› − 1

The so constructed function is bijective (∀) đ?‘ƒ, đ?‘„ ∈ [đ??ż]. It can be proved by the congruence of the triangles đ?‘ƒđ?‘‚đ?‘„ and đ?‘ƒâ€˛đ?‘‚′đ?‘„′ that ||đ?‘ƒđ?‘„|| = ||đ?‘ƒâ€˛đ?‘„′||, so [đ??ż] = [đ??żâ€˛]

â&#x;š if two convex polygons are decomposed

in the same number of triangles respectively congruent, they are congruent.

18

255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 13. 𝐿 = 𝑃1 𝑃2 … , 𝑃𝑛 ; 𝐿′ = 𝑃1′ 𝑃2′ … , 𝑃𝑛′ 𝐿~𝐿′ ⟹ (∃)𝐾 > 0 and 𝑓: 𝐿 → 𝐿′ such that ‖𝑃𝑄‖ = 𝑘‖𝑓(𝑃)𝑓(𝑄)‖ (∀)𝑃, 𝑄 ∈ 𝐿, and 𝑃𝐼′ = 𝑓(𝑃𝑖 ). Taking consecutively the peaks in the role of 𝑃 and 𝑄, we obtain: ‖𝑃1 𝑃2 ‖ =𝑘 ‖𝑃1′ 𝑃2′ ‖ ‖𝑃2 𝑃3 ‖ ‖𝑃2 𝑃3 ‖ = 𝑘‖𝑃2′ 𝑃3′ ‖ ⇒ ′ ′ = 𝑘 ‖𝑃2 𝑃3 ‖ ⟹ ⋮ ‖𝑃 ‖ 𝑃 𝑛−1 𝑛 ′ ‖𝑃𝑛−1 𝑃𝑛 ‖ = 𝑘‖𝑃𝑛−1 𝑃𝑛′ ‖ ⇒ ′ =𝑘 ‖𝑃𝑛−1 𝑃𝑛′ ‖ ‖𝑃𝑛 𝑃1 ‖ ‖𝑃𝑛 𝑃1 ‖ = 𝑘‖𝑃𝑛′ 𝑃1′ ‖ ⇒ ′ ′ = 𝑘 ‖𝑃𝑛 𝑃1 ‖ } ‖𝑃1 𝑃2 ‖ = 𝑘‖𝑃1′ 𝑃2′ ‖ ⇒

⟹𝑘=

‖𝑃1 𝑃2 ‖ ‖𝑃2 𝑃3 ‖ ‖𝑃1 𝑃2 ‖ + ‖𝑃2 𝑃3 ‖ + ⋯ + ‖𝑃𝑛−1 𝑃𝑛 ‖ + ‖𝑃𝑛 𝑃1 ‖ 𝑃 = = ⋯ = = . ′ ‖𝑃1′ 𝑃2′ ‖ ‖𝑃2′ 𝑃3′ ‖ ‖𝑃1′ 𝑃2′ ‖ + ‖𝑃2′ 𝑃3′ ‖ + ⋯ + ‖𝑃𝑛−1 𝑃𝑛′ ‖ + ‖𝑃𝑛′ 𝑃1′ ‖ 𝑃′

Solution to Problem 14.

𝜎[𝐴𝐷𝐶] =

2∙7 2

= 7; 𝜎[𝐴𝐵𝐶𝐷] = 2 ∙ 7 = 14 = 6‖𝐷𝐹‖ ⟹ ‖𝐷𝐹‖ =

Solution to Problem 15.

19

14 6

7

=3.

Florentin Smarandache

ℎ = 𝑏 ∙ sin 𝐶 ≤ 𝑏; 𝜎[𝐴𝐵𝐶] =

𝑎∙ℎ 2

is max. when ℎ is max.

max. ℎ = 𝑏 when sin𝐶 = 1 ⇒ 𝑚(𝐶) = 90 ⇒ 𝐴𝐵𝐶 has a right angle at 𝐶 .

Solution to Problem 16.

‖𝑀𝐷‖ = ‖𝐷𝐼‖ ⟹ 𝑀𝐷𝐼 – an isosceles triangle. ̂ ) = 𝑚(𝑀𝐼𝐷 ̂ ) = 450 ; ⟹ 𝑚(𝐷𝑀𝐼 ̂ ) = 𝑚(𝐴𝐹𝐿 ̂ ) = 𝑚(𝐵𝐸𝐻 ̂ ) = 𝑚(𝐸𝐻𝐵 ̂ ). The same way, 𝑚(𝐹𝐿𝐴 ‖𝑅𝐾‖ ⟹ ‖𝑆𝑃‖ = ‖𝑃𝑄‖ = ‖𝑄𝑅‖ = ‖𝑅𝑆‖ ⟹ 𝑆𝑅𝑄𝑃 is a square. ‖𝐴𝐵‖ = 𝑎, ‖𝐴𝐸‖ = 2‖𝑅𝐼‖2 =

𝑎2 9

2𝑎 , ‖𝑀𝐼‖ 3

=√

4𝑎 2 9

+

4𝑎 2 9

𝑎2

⟹ ‖𝑅𝐼‖2 = 18 ⟹ ‖𝑅𝐼‖ = 3

=

𝑎 √2

2𝑎√2 3

;

𝑎√2 6

;

=

2𝑎√2 𝑎 √2 𝑎√2 −2 6 = 3 ; 3 2𝑎 2 2 𝜎[𝑆𝑅𝑄𝑃] = 9 = 9 𝜎[𝐴𝐵𝐶𝐷].

‖𝑆𝑅‖ =

Solution to Problem 17. ‖𝐷𝐶‖ ∙ ‖𝐴𝐸‖ 2 ‖𝐷𝐶‖ ∙ ‖𝐵𝐹‖ ⟹ 𝜎[𝐴𝐶𝐷] = 𝜎[𝐵𝐶𝐷] 𝜎[𝐵𝐶𝐷] = 2 ‖𝐴𝐸‖ = ‖𝐵𝐹‖ } 𝜎[𝐴𝐶𝐷] =

20

255 Compiled and Solved Problems in Geometry and Trigonometry

𝜎[𝐴𝑂𝐷] = 𝜎[𝐴𝑀𝑂] + 𝜎[𝑀𝑂𝐷] ‖𝑀𝑂‖ ∙ ‖𝑂𝑃‖ ‖𝑂𝑀‖(‖𝑂𝑃‖ + ‖𝑂𝑄‖) ‖𝑂𝑀‖ ∙ ℎ 𝜎[𝐴𝑀𝑂] = 𝜎[𝑀𝑃𝑂] = ⟹ 𝜎[𝐴𝑂𝐷] = = 2 2 2 ‖𝑂𝑀‖ ∙ ‖𝑂𝑄‖ 𝜎[𝑀𝑂𝐷] = 𝜎[𝑀𝑂𝑄] = } 2 The same way, 𝜎[𝐵𝑂𝐶] =

‖𝑂𝑁‖ ∙ ℎ . 2

Therefore, 𝜎[𝐴𝑂𝐷] = 𝜎[𝐵𝑂𝐶] ⟹

‖𝑂𝑀‖ ∙ ℎ ‖𝑂𝑁‖ ∙ ℎ = ⟹ ‖𝑂𝑀‖ = ‖𝑂𝑁‖. 2 2

Solution to Problem 18. ‖𝐴𝐸‖ = ‖𝐸𝐷‖ ; We draw 𝑀𝑁 ⊥ 𝐴𝐵; 𝐷𝐶; ℎ

‖𝐸𝑁‖ = ‖𝐸𝑀‖ = ; 2 (‖𝐴𝐵‖ + ‖𝐷𝐶‖) ∙ ℎ ‖𝐴𝐵‖ ∙ ℎ ‖𝐷𝐶‖ ∙ ℎ (‖𝐴𝐵‖ + ‖𝐷𝐶‖) ∙ ℎ 1 𝜎[𝐵𝐸𝐶] = − − = = 𝜎[𝐴𝐵𝐶𝐷]; 2 4 4 4 2 [𝐴𝐵𝐶𝐷] Therefore, = 2𝜎[𝐵𝐸𝐶] .

Solution to Problem 19. 𝜎[𝐴𝐸𝐷𝑁′] is ct. because 𝐴, 𝐸, 𝐷, 𝑁′ are fixed points. Let a line through 𝐷, and we draw ∥ to sides 𝑁𝐷 and 𝐷𝐸. No matter how we draw a line through 𝐷, 𝜎[𝑄𝑃𝐴] is formed of: 𝜎[𝐴𝐸𝐷𝑁] + 𝜎[𝑁𝑃𝑂] + 𝜎[𝐷𝐸𝑄]. We have 𝜎[𝐴𝐸𝐷𝑁] constant in all triangles 𝑃𝐴𝑄.

21

Florentin Smarandache

Let’s analyse: ‖𝑁′𝐷‖ ∙ ℎ1 ‖𝐸𝑄′‖ ∙ ℎ2 ‖𝑁′𝐷‖ ‖𝐸𝑄‖ + = ∙ℎ ) (ℎ1 + ‖𝑁𝐷‖ 2 2 2 2 ‖𝑁′𝐷‖ ‖𝑁 ′ 𝐷‖ ℎ2 [(ℎ1 − ℎ2 )2 + 2ℎ1 ℎ2 ]. = ∙ (ℎ1 + ∙ ℎ2 ) = 2 ℎ1 2ℎ1

𝜎[𝑃𝑁 ′ 𝐷] + 𝜎[𝐷𝐸𝑄] =

∆𝐴𝑀𝑁 is minimal when ℎ1 = ℎ2 ⟹ 𝐷 is in the middle of |𝑃𝑄|. The construction is thus: ∆𝐴𝑁𝑀 where 𝑁𝑀 || 𝐸𝑁′. In this case we have |𝑁𝐷| ≡ |𝐷𝑀|.

Solution to Problem 20.

𝜎[𝐴𝐵𝐶] =

‖𝐵𝐶‖ ∙ ‖𝐴𝐴′‖ 2

Let the median be |𝐴𝐸|, and 𝑃 be the centroid of the triangle. Let 𝑃𝐷 ⊥ 𝐵𝐶. 𝜎[𝐵𝑃𝐶] =

‖𝐵𝐶‖∙‖𝑃𝐷‖ 2

.

′ 𝐴𝐴′ ⊥ 𝐵𝐶 } ⇒ 𝐴𝐴′ ∥ 𝑃𝐷 ⇒ ∆𝑃𝐷𝐸~∆𝐴𝐴′ 𝐸 ⇒ ‖𝑃𝐷‖ = ‖𝑃𝐸‖ = 1 ⇒ ‖𝑃𝐷‖ = ‖𝐴𝐴 ‖ ⇒ 𝜎[𝐵𝑃𝐶] ‖𝐴𝐴′ ‖ ‖𝐴𝐸‖ 3 𝑃𝐷 ⊥ 𝐵𝐶 3 ‖𝐴𝐴′‖ ′ ‖𝐵𝐶‖ ∙ 3 = 1 ‖𝐵𝐶‖ ∙ ‖𝐴𝐴 ‖ = 1 𝜎[𝐴𝐵𝐶]. = 2 3 2 3 1 We prove in the same way that 𝜎[𝑃𝐴𝐶] = 𝜎[𝑃𝐴𝐵] = 3 𝜎[𝐴𝐵𝐶], so the specific

point is the centroid. 22

255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 21. Let 𝑀, 𝑁 ∈ 𝐴𝐵 such that 𝑀 ∈ |𝐴𝑁|. We take 𝑀𝑀′ ∥ 𝐵𝐶, 𝑀𝑁′ ∥ 𝐵𝐶.

∆𝐴𝑀𝑀′ ~∆𝐴𝐵𝐶 ⇒

𝜎[𝐴𝑀𝑀′ ] 𝐴𝑀 =( ) 𝜎[𝐴𝐵𝐶] 𝐴𝐵

2

‖𝐴𝑀‖ ‖𝐴𝐵‖ 1 1 𝜎[𝐴𝑀𝑀′] = 𝜎, ( ; ∆𝐴𝑁𝑁′~∆𝐴𝐵𝐶 ⟹ ) = , ‖𝐴𝑀‖ = ‖𝐴𝐵‖ 3 3 √3 2

‖𝐴𝑁‖ 𝜎[𝐴𝑁𝑁′] 2 = ( ) ‖𝐴𝑁‖ 2 2 ‖𝐴𝐵‖ 𝜎[𝐴𝐵𝐶] ⟹ ( ) = ⟹ ‖𝐴𝑁‖ = √ ‖𝐴𝐵‖ . ‖𝐴𝐵‖ 3 3 2 𝜎[𝐴𝑁𝑁′] = 𝜎[𝐴𝐵𝐶] 3 }

Solution to Problem 22.

‖𝑂𝐷‖ = 𝑎, ‖𝑂𝐴‖ = 𝑏 ; 1

𝜎[∆𝐶𝑀′𝑀] = 𝜎[𝑀𝑀′𝑁′𝑁] = 𝜎[𝑁𝑁′𝐵𝐴] = 3 𝜎[𝐴𝐵𝐶𝐷] ; 𝜎[𝑂𝐷𝐶] ‖𝑂𝐷‖2 𝑎 𝜎[𝑂𝐷𝐶] 𝑎2 𝜎[𝑂𝐷𝐶] = = ⟹ = ⟹ 2 2 2 𝜎[𝑂𝐴𝐵] ‖𝑂𝐴‖ 𝑏 𝜎[𝑂𝐴𝐵] − 𝜎[𝑂𝐷𝐶] 𝑏 − 𝑎 𝜎[𝐴𝐵𝐶𝐷] 𝑎2 (1) = 2 𝑏 − 𝑎2

∆𝑂𝐷𝐶~∆𝑂𝐴𝐵 ⟹

23

Florentin Smarandache 2

‖𝑂𝐷‖ 𝜎[𝑂𝐷𝐶] 𝜎[𝑂𝐷𝐶] 𝑎2 ∆𝑂𝐷𝐶~∆𝑂𝑀𝑀 ⟹ =( = ) ⟹ ‖𝑂𝑀‖ 𝜎[𝑂𝑀𝑀′ ] 𝜎[𝑂𝑀𝑀′ ] − 𝜎[𝑂𝐷𝐶] ‖𝑂𝑀‖2 𝜎[𝑂𝐷𝐶] 𝑎2 𝜎[𝑂𝐷𝐶] 𝑎2 (2) ⟹ = ⟹ = 2 − 𝑎2 1 ‖𝑂𝑀‖ 𝜎[𝐷𝐶𝑀𝑀′ ] ‖𝑂𝑀‖2 − 𝑎2 3 𝜎[𝐴𝐵𝐶𝐷] ′

‖𝑂𝐷‖ 𝜎[𝑂𝐷𝐶] 𝑎2 𝜎[𝐼𝐷𝐶] 𝑎2 = = ⟹ = 𝜎[𝐷𝑁𝑁 ′ ] ‖𝑂𝑁‖ ‖𝑂𝑁‖ 𝜎[𝑂𝑁𝑁′] − 𝜎[𝑂𝐷𝐶] ‖𝑂𝑁‖2 𝜎[𝑂𝐷𝐶] 𝑎2 𝜎[𝑂𝐷𝐶] 𝑎2 (3) ⟹ = ⟹ = 2 ‖𝑂𝑁‖2 − 𝑎2 𝜎[𝐷𝐶𝑁𝑁′] ‖𝑂𝑁‖2 − 𝑎2 𝜎[𝐴𝐵𝐶𝐷] 3

∆𝑂𝑁𝑁 ′ ~∆𝑂𝐷𝐶 ⟹

We divide (1) by (3): 2 ‖𝑂𝑁‖2 − 𝑎2 𝑎2 + 2𝑏 2 2 2 2 2 2 ‖𝑂𝑁‖ = ⟹ 3‖𝑂𝑁‖ − 3𝑎 = 2𝑏 − 2𝑎 ⟹ = . 3 𝑏 2 − 𝑎2 3

Solution to Problem 23. ‖𝑂𝐴‖ = 𝑟 → ‖𝐷𝐸‖ = 2𝑟; 𝜎hexagon =

3𝑟 2 √2 2

(1)

𝐷𝐸𝐹𝑂 a square inscribed in the circle with radius 𝑅 ⟹ ⟹ 𝑙4 = 𝑅√2 = ‖𝐷𝐸‖ ⟹ 𝑃√2 = 2𝑟 ⟹ 𝑅 = 𝑟√2 ‖𝑂𝑀‖ = 𝑅 = 𝑟√2 √3 ‖𝑂𝑀‖ ∙ ‖𝑂𝑁‖ sin 120 𝑟√2 ∙ 𝑟√2 ∙ 2 𝑟 2 √3 𝜎[𝑂𝑀𝑁] = = = 2 2 2 𝜎[𝑀𝑁𝑃] = 3𝜎[𝑂𝑀𝑁] = 3

𝑟 2 √3 3𝑟 2 √2 (2) = 2 2

From (1) and (2) ⟹ 𝜎[𝑀𝑁𝑃] = 𝜎hexagon .

24

255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 24.

‖𝐴𝐵‖2 = ‖𝐵𝐶‖ ∙ ‖𝐵𝐴′‖ We construct the squares 𝐵𝐶𝐸𝐷 on the hypotenuse and 𝐴𝐵𝐹𝐺 on the leg. We draw 𝐴𝐴′ ⊥ 𝐵𝐶. 𝜎[𝐴𝐵𝐹𝐺] = ‖𝐴𝐵‖2 𝜎[𝐴′𝐵𝐷𝐻] = ‖𝐵𝐷‖ ∙ ‖𝐵𝐴′ ‖ = ‖𝐵𝐶‖ ∙ ‖𝐵𝐴′ ‖ …

Solution to Problem 25.

𝜎(𝑠1 ) = 𝜎[𝐴𝐵𝐶] − 3𝜎[sect. 𝐴𝐷𝐻] 𝑙 2 √3 (2𝑎)2 √3 𝜎[𝐴𝐵𝐶] = = = 𝑎2 √3 4 4 𝑟2 ̂) 𝜎[sect. 𝐴𝐷𝐻] = 𝑚(𝐷𝐻 2 𝜋 𝜋 𝜋 ̂)= ̂)= 𝑚(𝐷𝐻 𝑚(𝐷𝐻 ∙ 600 = 180 180 3 𝑎2 𝜋 𝜋𝑎2 𝜎[sect. 𝐴𝐷𝐻] = ∙ = 2 3 2 2 𝜋𝑎 𝜋𝑎2 (1) 𝜎(𝑠1 ) = 𝑎2 √3 − 3 = 𝑎2 √3 − 6 2

25

Florentin Smarandache

𝜎(𝑠2 ) = 𝜎[sect. 𝐴𝐸𝐺] − 𝜎[sect. 𝐴𝐵𝐶] − 𝜎[sect. 𝐸𝐶𝐹] − 𝜎[sect. 𝐺𝐵𝐹] (3𝑎)2 𝜋 𝑎2 𝜋 9𝑎2 𝜋 𝜋𝑎2 𝜋𝑎2 = ∙ ∙ 60 − 𝑎2 √3 − ∙ ∙ 120 = ∙ − 𝑎2 √3 − − 2 180 2 180 2 3 3 3 3𝜋𝑎2 2𝜋𝑎2 = − − 𝑎2 √3 (2) 2 3 From (1) and (2) ⟹ 𝜎(𝑠1 ) + 𝜎(𝑠2 ) =

3𝜋𝑎2 2𝜋𝑎2 𝜋𝑎2 2𝜋𝑎2 𝜋𝑎2 − − = = . 2 3 2 6 3

Solution to Problem 26.

‖𝐴𝐷‖2 = 𝑟22 − 𝑟12 𝜎[𝐿(𝑂, 𝑟1 )] = 𝜋𝑟12 𝜎[𝐿(𝑂, 𝑟2 )] = 𝜋𝑟22 𝜎[annulus] = 𝜋𝑟22 − 𝜋𝑟12 = 𝜋(𝑟22 − 𝑟12 ) (1) 𝜎[disk diameter‖𝐴𝐵‖] = 𝜋‖𝐴𝐷‖2 = 𝜋(𝑟22 − 𝑟12 ) (2) From (1) and (2) ⟹ 𝜎[annulus] = 𝜎[disk diameter].

Solution to Problem 27.

26

255 Compiled and Solved Problems in Geometry and Trigonometry

𝜎[𝐶𝐷𝐷′𝐶′] 2 𝜎[𝐶𝐷𝐷′𝐶′] = 𝜎seg [𝐶𝐷𝐵𝐷′𝐶′] − 𝜎seg [𝐷𝐵𝐷′] 𝜎[𝐶𝐷𝐸𝐹] =

̂ ) = 𝑚(𝐵𝐷 ̂ ) = 𝜋 − 2𝑎 ̂ ) = 𝑎 ⟹ 𝑚(𝐶𝐷 We denote 𝑚(𝐴𝐶 2 𝑟2 𝜎sect. = (𝛼 − sin 𝛼) 2 2 𝑟 𝑟2 𝜎[𝐶𝐷𝐵𝐷′𝐶′] = [𝜋 − 2𝛼 − sin(𝜋 − 2𝛼)] = 𝑞 [𝜋 − 2𝛼 − sin(𝜋 − 2𝛼)] 2 2 𝑟2 𝜎𝐴 = (2𝛼 − sin 2𝛼) 2 𝑟2 𝑟2 𝜎[𝐶𝐷𝐷′𝐶′] = 𝜎[𝐶𝐷𝐵𝐷′𝐶′] − 𝜎[𝐷𝐵𝐷 ′ ] = (𝜋 − 2𝛼 − sin 2𝛼) = (2𝛼 − sin 2𝛼) 2 2 𝑟2 𝑟2 = (𝜋 − 2𝛼 − sin 2𝛼 − 2𝛼 + sin 2𝛼) = (𝜋 − 4𝛼) 2 2 𝜎[𝐶𝐷𝐷 ′ 𝐶 ′ ] 𝑟 2 𝑟2 𝜋 ⟹ 𝜎[𝐶𝐷𝐸𝐹] = = (𝜋 − 4𝛼) = ( − 2𝛼) (1) 2 4 2 2 𝑟2 𝑟2 𝜋 ̂ ) = ( − 2𝛼) (2) 𝜎[sect. 𝐶𝑂𝐷] = 𝑚(𝐶𝐷 2 2 2 From (1) and (2) ⟹ 𝜎[𝐶𝐷𝐸𝐹] = 𝜎[sect. 𝐶𝑂𝐷]. ‖𝑂1 𝐹‖ = ‖𝑂𝐸‖ 𝜎[square] = ‖𝐷𝐸‖2 = ‖𝑂𝐴‖2 =

𝑏𝑐 = 𝑉[𝐴𝐵𝐶] 2

Solution to Problem 28.

̂) = 𝜇(𝐴𝑂𝐵

𝜎[𝐴𝑂𝐵] =

𝜋 𝑟 2 sin 4 2

𝜎[orthogon] = 8 ∙

27

𝜋 4

√2 𝑟2 2 𝑟 2 √2 = = 2 4 𝑟 2 √2 = 2√2𝑟 2 4

Florentin Smarandache

Solution to Problem 29.

đ?œŽ[đ??´đ??ľđ??ś] = đ?œŽ[đ??´đ?‘€đ??ľ] + đ?œŽ[đ??´đ?‘€đ??ś] + đ?œŽ[đ?‘€đ??ľđ??ś] â&#x;š đ?‘Žâ„Žđ?‘Ž = đ?‘Žđ?‘‘3 + đ?‘Žđ?‘‘2 + đ?‘Žđ?‘‘1 đ?‘‘1 + đ?‘‘2 + đ?‘‘3 = â„Žđ?‘Ž (đ?‘Ž is the side of equilateral triangle) â&#x;š đ?‘‘1 + đ?‘‘2 + đ?‘‘3 =

đ?‘Žâˆš3 2

(because â„Žđ?‘Ž =

đ?‘Žâˆš3 2

).

Solution to Problem 30.

đ??´đ??ľđ??ś − given ∆ â&#x;š đ?‘Ž, đ?‘?, đ?‘?, â„Ž − constant đ?‘Žâ„Ž đ?œŽ[đ??´đ??ľđ??ś] = 2 đ?œŽ[đ??´đ??ľđ??ś] = đ?œŽ[đ??´đ?‘€đ??ľ] + đ?œŽ[đ??´đ?‘€đ??ś] đ?‘Žâ„Ž đ?‘?đ?‘Ľ đ?‘?đ?‘Ś đ?‘? đ?‘? â&#x;š = + â&#x;š đ?‘?đ?‘Ľ + đ?‘?đ?‘Ś = đ?‘Žđ?‘? â&#x;š đ?‘Ľ+ đ?‘Ś = 1 â&#x;š đ?‘˜đ?‘Ľ + đ?‘™đ?‘Ś = 1, 2 2 2 đ?‘Žâ„Ž đ?‘Žâ„Ž đ?‘? đ?‘? where đ?‘˜ = đ?‘Žâ„Ž and đ?‘™ = đ?‘Žâ„Ž .

Solution to Problem 31.

28

255 Compiled and Solved Problems in Geometry and Trigonometry

We draw đ??´đ??´â€˛ ⊼ đ??ľđ??ś; đ?‘ đ?‘ ′ ⊼ đ??ľđ??ś; đ??ˇđ??ˇâ€˛ ⊼ đ??ľđ??ś ⇒ the trapezoid đ??´đ??´â€˛ đ??ˇ ′ đ??ˇ â&#x;š ‖đ?‘ đ?‘ ′ ‖ =

đ??´đ??´â€˛ âˆĽ đ?‘ đ?‘ ′ âˆĽ đ??ˇđ??ˇ ′ } â&#x;š đ?‘€đ?‘ ′ median line in ‖đ??´đ?‘ ‖ = ‖đ?‘ đ??ˇâ€–

‖đ??´đ??´â€˛â€–+‖đ??ˇđ??ˇâ€˛â€–

‖đ??ľđ??śâ€–+‖đ?‘ đ?‘ ′‖

2

2

, đ?œŽ[đ??ľđ??śđ?‘ ] =

.

Solution to Problem 32.

First, we construct a quadrilateral with the same area as the given pentagon. We draw through C a parallel to BD and extend |AB| until it intersects the parallel at M. đ?œŽ[đ??´đ??ľđ??śđ??ˇđ??¸] = đ?œŽ[đ??´đ??ľđ??ˇđ??¸] + đ?œŽ[đ??ľđ??śđ??ˇ], đ?œŽ[đ??ľđ??śđ??ˇ] = đ?œŽ[đ??ľđ??ˇđ?‘€] (have the vertices on a parallel at the base). Therefore, đ?œŽ[đ??´đ??ľđ??śđ??ˇđ??¸] = đ?œŽ[đ??´đ?‘€đ??ˇđ??¸]. Then, we consider a triangle with the same area as the quadrilateral đ??´đ?‘€đ??ˇđ??¸. We draw a parallel to đ??´đ??ˇ, đ?‘ is an element of the intersection with the same parallel. đ?œŽ[đ??´đ?‘€đ??ˇđ??¸] = đ?œŽ[đ??´đ??ˇđ??¸] + đ?œŽ[đ??´đ??ˇđ??¸] = đ?œŽ[đ??´đ??ˇđ??¸] + đ?œŽ[đ??´đ??ˇđ?‘ ] = đ?œŽ[đ??¸đ??ˇđ?‘ ].

29

Florentin Smarandache

Solution to Problem 33.

‖đ??´đ??¸â€– = ‖đ??¸đ??śâ€– ‖đ??¸đ??šâ€–đ??ľđ??ˇ â&#x;š đ?œŽ[đ??ľđ??ˇđ??š] = đ?œŽ[đ??ľđ??ˇđ??¸] đ?œŽ[đ??´đ??ľđ??šđ??ˇ] = đ?œŽ[đ??´đ??ľđ??¸đ??ˇ] (1) đ?œŽ[đ??´đ??ˇđ??¸] = đ?œŽ[đ??ˇđ??¸đ??ś] equal bases and the same height; đ?œŽ[đ??´đ??ˇđ??¸] = đ?œŽ[đ??ˇđ??¸đ??ś] đ?œŽ[đ??´đ??ľđ??¸đ??ˇ] = đ?œŽ[đ??ľđ??¸đ??ˇđ??ś] (2) đ?œŽ[đ??ˇđ??¸đ??š] = đ?œŽ[đ??ľđ??¸đ??š] the same base and the vertices on parallel lines at the base;

30

255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 34.

It is proved in the same way that:

â&#x;š đ?‘€đ?‘ đ?‘ƒđ?‘„ rhombus with right angleâ&#x;š đ?‘€đ?‘ đ?‘ƒđ?‘„ is a square.

It is proved in the same way that all the peaks of the octagon are elements of the axis of symmetry of the square, thus the octagon is regular.

31

Florentin Smarandache

Consider the square separately.

Solution to Problem 35.

‖đ?‘‚đ?‘€â€– = ‖đ?‘ đ?‘€â€– = ‖đ?‘ đ??ľâ€– ‖đ??ˇđ??śâ€–? â&#x;š ∆đ?‘€đ?‘‚đ??ś = ∆đ?‘ đ??ľđ??´ â&#x;š ‖đ?‘€đ??śâ€– = ‖đ??´đ?‘ ‖ It is proved in the same way that ∆đ??ˇđ??´đ?‘€ = ∆đ??ľđ??śđ?‘ â&#x;š ‖đ?‘€đ??śâ€– = ‖đ?‘ đ??śâ€–. Thus đ??´đ?‘ đ??śđ?‘€ is a parallelogram. 32

255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 36.

To determine the angle Îą:

We write

thus we have established the positions of the lines of the locus.

constant for đ??´, đ??ľ, đ??ś, đ??ˇ – fixed points. We must find the geometrical locus of points đ?‘€ such that the ratio of the distances from this point to two concurrent lines to be constant.

33

Florentin Smarandache ‖đ?‘€đ??¸â€– ‖đ?‘€đ??šâ€–

= đ?‘˜. Let M' be another point with the same property, namely

‖đ?‘€â€˛đ??¸â€˛â€– ‖đ?‘€â€˛đ??šâ€˛â€–

= đ?‘˜.

đ?‘ƒ, đ?‘€, đ?‘€â€˛ collinear â&#x;š the locus is a line that passes through đ?‘ƒ. When the points are in âˆ˘đ??śđ?‘ƒđ??ľ we obtain one more line that passes through đ?‘ƒ. Thus the locus is formed by two concurrent lines through đ?‘ƒ, from which we eliminate point đ?‘ƒ, because the distances from đ?‘ƒ to both lines are 0 and their ratio is indefinite. Vice versa, if points đ?‘ and đ?‘ ′ are on the same line passing through đ?‘ƒ, the ratio of their distances to lines đ??´đ??ľ and đ??śđ??ˇ is constant.

Solution to Problem 37. We show in the same way as in the previous problem that: ‖đ?‘€đ??¸â€– ‖đ?‘€đ??¸â€– đ?‘˜ đ?‘˜đ?‘‘ =đ?‘˜â&#x;š = â&#x;š ‖đ?‘€đ??¸â€– = , ‖đ?‘€đ??šâ€– ‖đ?‘€đ??šâ€– + ‖đ?‘€đ??¸â€– 1 − đ?‘˜ 1+đ?‘˜ and the locus of the points which are located at a constant distance from a given line is a parallel to the respective line, located between the two parallels. If ||đ??´đ??ľ|| > ||đ??śđ??ˇ|| â&#x;š đ?‘‘(đ?‘€đ??´đ??¸) < đ?‘‘(đ?‘€đ??śđ??ˇ). 34

255 Compiled and Solved Problems in Geometry and Trigonometry

Then, if đ?‘€đ??¸ đ?‘€đ??¸ đ?‘˜ đ?‘€đ??¸ đ?‘˜ đ?‘˜đ?‘‘ =đ?‘˜â&#x;š = â&#x;š = â&#x;š đ?‘€đ??¸ = , đ?‘€đ??š đ?‘€đ??š − đ?‘€đ??¸ 1 − đ?‘˜ đ?‘‘ 1−đ?‘˜ 1−đ?‘˜ thus we obtain one more parallel to đ??´đ??ľ.

Solution to Problem 38. Solution no. 1

We suppose that ABCD is not a parallelogram. Let {đ??ź} = đ??´đ??ľ ∊ đ??śđ??ˇ. We build đ??¸ ∈ (đ??źđ??´ such that đ??źđ??¸ = đ??´đ??ľ and đ??š ∈ (đ??źđ??ś such that đ??źđ??š = đ??śđ??ˇ. If đ?‘€ a point that verifies đ?œŽ[đ??´đ??ľđ?‘€] + đ?œŽ[đ??śđ??ˇđ?‘€] = 1 (1), then, because đ?œŽ[đ??´đ??ľđ?‘€] = đ?œŽ[đ?‘€đ??źđ??¸] and đ?œŽ[đ??śđ??ˇđ?‘€] = đ?œŽ[đ?‘€đ??źđ??š], it results that đ?œŽ[đ?‘€đ??źđ??¸] + đ?œŽ[đ?‘€đ??źđ??š] = đ?‘˜ (2). We obtain that đ?œŽ[đ?‘€đ??¸đ??źđ??š] = đ?‘˜. On the other hand, the points đ??¸, đ??š are fixed, therefore đ?œŽ[đ??źđ??¸đ??š] = đ?‘˜ ′ = const. That is, đ?œŽ[đ?‘€đ??¸đ??š] = đ?‘˜ − đ?‘˜ ′ = const. Because đ??¸đ??š = const., we have đ?‘‘(đ?‘€, đ??¸đ??š) =

2(đ?‘˜âˆ’đ?‘˜ ′ ) đ??¸đ??š

= const., which shows that đ?‘€

belongs to a line that is parallel to đ??¸đ??š, taken at the distance

2(đ?‘˜âˆ’đ?‘˜ ′ ) đ??¸đ??š

.

Therefore, the locus points are those on the line parallel to đ??¸đ??š, located inside the quadrilateral đ??´đ??ľđ??śđ??ˇ. They belong to the segment [đ??¸â€˛đ??šâ€˛] in Fig. 1. Reciprocally, if đ?‘€ ∈ [đ??¸â€˛đ??šâ€˛], then đ?œŽ[đ?‘€đ??´đ??ľ] + đ?œŽ[đ?‘€đ??śđ??ˇ] = đ?œŽ[đ?‘€đ??źđ??¸] + đ?œŽ[đ?‘€đ??źđ??š] = đ?œŽ[đ?‘€đ??¸đ??źđ??š] = đ?œŽ[đ??źđ??¸đ??š] + đ?œŽ[đ?‘€đ??¸đ?‘ƒ] = đ?‘˜ ′ +

đ??¸đ??šâˆ™2(đ?‘˜âˆ’đ?‘˜ ′ ) 2∙đ??¸đ??š

35

= đ?‘˜.

Florentin Smarandache

In conclusion, the locus of points đ?‘€ inside the quadrilateral đ??´đ??ľđ??śđ??ˇ which occurs for relation (1) where đ?‘˜ is a positive constant smaller than đ?‘† = đ?œŽ[đ??´đ??ľđ??śđ??ˇ] is a line segment. If đ??´đ??ľđ??śđ??ˇ is a trapeze having đ??´đ??ľ and đ??śđ??ˇ as bases, then we reconstruct the reasoning as đ??´đ??ˇ ∊ đ??ľđ??ś = {đ??ź} and đ?œŽ[đ?‘€đ??´đ??ˇ] + đ?œŽ[đ?‘€đ??ľđ??ś] = đ?‘ − đ?‘˜ = const. If đ??´đ??ľđ??śđ??ˇ is a parallelogram, one shows without difficulty that the locus is a segment parallel to đ??´đ??ľ.

Solution no. 2 (Ion Patrascu) We prove that the locus of points đ?‘€ which verify the relationship đ?œŽ[đ?‘€đ??´đ??ľ] + đ?œŽ[đ?‘€đ??śđ??ˇ] = đ?‘˜ (1) from inside the convex quadrilateral đ??´đ??ľđ??śđ??ˇ of area đ?‘ (đ?‘˜ ⊂ đ?‘ ) is a line segment. Let’s suppose that đ??´đ??ľ ∊ đ??śđ??ˇ = {đ??ź}, see Fig. 2. There is a point đ?‘ƒ of the locus which 2đ?‘˜

belongs to the line đ??śđ??ˇ. Therefore, we have (đ?‘ƒ; đ??´đ??ľ) = đ??´đ??ľ . Also, there is the point đ?‘„ ∈ đ??´đ??ľ such that đ?‘‘(đ?‘„; đ??śđ??ˇ) =

2đ?‘˜ . đ??śđ??ˇ

Now, we prove that the points from inside the quadrilateral đ??´đ??ľđ??śđ??ˇ that are on the segment [đ?‘ƒđ?‘„] belong to the locus.

Let đ?‘€ ∈ int[đ??´đ??ľđ??śđ??ˇ] ∊ [đ?‘ƒđ?‘„]. We denote đ?‘€1 and đ?‘€2 the projections of đ?‘€ on đ??´đ??ľ and đ??śđ??ˇ respectively. Also, let đ?‘ƒ1 be the projection of đ?‘ƒ on đ??´đ??ľ and đ?‘„1 the projection of đ?‘„ on đ??śđ??ˇ. The triangles đ?‘ƒđ?‘„đ?‘„1 and đ?‘ƒđ?‘€đ?‘€2 are alike, which means that đ?‘€đ?‘€2 đ?‘€đ?‘ƒ (2), = đ?‘„đ?‘„1 đ?‘ƒđ?‘„

36

255 Compiled and Solved Problems in Geometry and Trigonometry

and the triangles đ?‘€đ?‘€1 đ?‘„ and đ?‘ƒđ?‘ƒ1 đ?‘„ are alike, which means that đ?‘€đ?‘€1 đ?‘€đ?‘„ (3). = đ?‘ƒđ?‘ƒ1 đ?‘ƒđ?‘„ By adding member by member the relations (2) and (3), we obtain đ?‘€đ?‘€2 đ?‘€đ?‘€1 đ?‘€đ?‘ƒ + đ?‘€đ?‘„ + = = 1 (4). đ?‘„đ?‘„1 đ?‘ƒđ?‘ƒ1 đ?‘ƒđ?‘„ 2đ?‘˜

2đ?‘˜

Substituting in (4), đ?‘„đ?‘„1 = đ??śđ??ˇ and đ?‘ƒ1 = đ??´đ??ľ , we get đ??´đ??ľ ∙ đ?‘€đ?‘€1 + đ??śđ??ˇ ∙ đ?‘€đ?‘€2 = 2đ?‘˜, that is đ?œŽ[đ?‘€đ??´đ??ľ] + đ?œŽ[đ?‘€đ??śđ??ˇ] = đ?‘˜. We prove now by reductio ad absurdum that there is no point inside the quadrilateral đ??´đ??ľđ??śđ??ˇ that is not situated on the segment [đ?‘ƒđ?‘„], built as shown, to verify the relation (1). Let a point đ?‘€â€˛ inside the quadrilateral đ??´đ??ľđ??śđ??ˇ that verifies the relation (1), đ?‘€â€˛ ∉ [đ?‘ƒđ?‘„]. We build đ?‘€â€˛ đ?‘‡ ∊ đ??´đ??ľ, đ?‘€â€˛ đ?‘ˆ âˆĽ đ??śđ??ˇ, where đ?‘‡ and đ?‘ˆ are situated on [đ?‘ƒđ?‘„], see Fig. 3.

We denote đ?‘€1′ , đ?‘‡1 , đ?‘ˆ1 the projections of đ?‘€1 , đ?‘‡, đ?‘ˆ on đ??´đ??ľ and đ?‘€2′ , đ?‘‡2 , đ?‘ˆ2 the projections of the same points on đ??śđ??ˇ. We have the relations: đ?‘€â€˛ đ?‘€1′ ∙ đ??´đ??ľ + đ?‘€â€˛ đ?‘€2′ ∙ đ??śđ??ˇ = 2đ?‘˜ (5), đ?‘‡đ?‘‡1 ∙ đ??´đ??ľ + đ?‘‡đ?‘‡2 ∙ đ??śđ??ˇ = 2đ?‘˜ (6). Because đ?‘€â€˛ đ?‘€1′ = đ?‘‡đ?‘‡1 and đ?‘€â€˛ đ?‘€2′ = đ?‘ˆđ?‘ˆ2 , substituting in (5), we get: đ?‘‡đ?‘‡1 ∙ đ??´đ??ľ + đ?‘ˆđ?‘ˆ2 ∙ đ??śđ??ˇ = 2đ?‘˜ (7). From (6) and (7), we get that đ?‘‡đ?‘‡2 = đ?‘ˆđ?‘ˆ2 , which drives us to đ?‘ƒđ?‘„ âˆĽ đ??śđ??ˇ, false!

37

Florentin Smarandache

Problems in Geometry and Trigonometry 39. Find the locus of the points such that the sum of the distances to two concurrent lines to be constant and equal to đ?‘™. Solution to Problem 39

40. Show that in any triangle đ??´đ??ľđ??ś we have: a. đ?‘? đ?‘?đ?‘œđ?‘ đ??ś + đ?‘? đ?‘?đ?‘œđ?‘ đ??ľ = đ?‘Ž; b. đ?‘? đ?‘?đ?‘œđ?‘ đ??ľ + đ?‘? đ?‘?đ?‘œđ?‘ đ??ś = đ?‘Ž đ?‘?đ?‘œđ?‘ (đ??ľ − đ??ś). Solution to Problem 40

41. Show that among the angles of the triangle đ??´đ??ľđ??ś we have: a. đ?‘? đ?‘?đ?‘œđ?‘ đ??ś − đ?‘? đ?‘?đ?‘œđ?‘ đ??ľ =

đ?‘? 2 −đ?‘Ž2 đ?‘Ž

;

b. 2(đ?‘?đ?‘? đ?‘?đ?‘œđ?‘ đ??´ + đ?‘Žđ?‘? đ?‘?đ?‘œđ?‘ đ??ľ + đ?‘Žđ?‘? đ?‘?đ?‘œđ?‘ đ??ś = đ?‘Ž2 + đ?‘? 2 + đ?‘? 2 . Solution to Problem 41

2

42. Using the law of cosines prove that 4đ?‘š đ?‘Ž = 2(đ?‘? 2 + đ?‘? 2 ) − đ?‘Ž2 , where đ?‘šđ?‘Ž is the length of the median corresponding to the side of đ?‘Ž length. Solution to Problem 42

43. Show that the triangle đ??´đ??ľđ??ś where

đ?‘Ž+đ?‘? đ?‘?

đ??ľ

= cot 2 is right-angled. Solution to Problem 43

44. Show that, if in the triangle đ??´đ??ľđ??ś we have cot đ??´ + cot đ??ľ = 2 cot đ??ś ⇒ đ?‘Ž2 + đ?‘? 2 = 2đ?‘? 2 . Solution to Problem 44

45. Determine the unknown elements of the triangle đ??´đ??ľđ??ś, given: a. đ??´, đ??ľ and đ?‘?; b. đ?‘Ž + đ?‘? = đ?‘š, đ??´ and đ??ľ; c. đ?‘Ž, đ??´; đ?‘? − đ?‘? = đ?‘Ž. Solution to Problem 45

38

255 Compiled and Solved Problems in Geometry and Trigonometry

46. Show that in any triangle đ??´đ??ľđ??ś we have tan

đ??´âˆ’đ??ľ 2

tan 2 = đ?‘Ž+đ?‘? (tangents đ??ś

đ?‘Žâˆ’đ?‘?

theorem). Solution to Problem 46

47. In triangle đ??´đ??ľđ??ś it is given đ??´Ě‚ = 60° and

đ?‘? đ?‘?

= 2 + √3. Find tan

đ??ľâˆ’đ??ś 2

and angles

đ??ľ and đ??ś. Solution to Problem 47

48. In a convex quadrilateral đ??´đ??ľđ??śđ??ˇ, there are given ‖đ??´đ??ˇâ€– = 7(√6 − √2), ‖đ??śđ??ˇâ€– = 33

đ?œ‹

5

13, ‖đ??ľđ??śâ€– = 15, đ??ś = arccos 65, and đ??ˇ = 4 + arccos 13 . The other angles of the quadrilateral and ‖đ??´đ??ľâ€– are required. Solution to Problem 48

49. Find the area of ∆đ??´đ??ľđ??ś when: a. đ?‘Ž = 17, đ??ľ = arcsin

24 25

, đ??ś = arcsin

12

;

13

b. đ?‘? = 2, đ??´Ě‚ ∈ 135°, đ??śĚ‚ ∈ 30°; c. đ?‘Ž = 7, đ?‘? = 5, đ?‘? = 6; d. đ??´Ě‚ ∈ 18°, đ?‘? = 4, đ?‘? = 6. Solution to Problem 49

50. How many distinct triangles from the point of view of symmetry are there such that đ?‘Ž = 15, đ?‘? = 13, đ?‘ = 24? Solution to Problem 50

51. Find the area of ∆đ??´đ??ľđ??ś if đ?‘Ž = √6, đ??´Ě‚ ∈ 60°, đ?‘? + đ?‘? = 3 + √3. Solution to Problem 51

52. Find the area of the quadrilateral from problem 48. Solution to Problem 52

39

Florentin Smarandache

53. If �� is the area of the regular polygon with � sides, find: �3 , �4 , �6 , �8 , �12 , �20 in relation to �, the radius of the circle inscribed in the polygon. Solution to Problem 53

54. Find the area of the regular polygon đ??´đ??ľđ??śđ??ˇ ‌ đ?‘€ inscribed in the circle with radius đ?‘…, knowing that:

1 ‖đ??´đ??ľâ€–

1

1

= ‖đ??´đ??śâ€– + ‖đ??´đ??ˇâ€– . Solution to Problem 54

55. Prove that in any triangle ABC we have: đ??´

a. đ?‘&#x; = (đ?‘? − đ?‘Ž) tan 2 ; đ??´

b. đ?‘† = (đ?‘? − đ?‘Ž) tan 2 ; đ??´

đ??ľ

đ??ś

c. đ?‘? = 4đ?‘… cos 2 cos 2 cos 2; đ??´

đ??ľ

đ??ś

d. đ?‘? − đ?‘Ž = 4đ?‘… cos 2 cos 2 cos 2; e. đ?‘šđ?‘Ž2 = đ?‘… 2 (sin2 đ??´ + 4 cos đ??´ sin đ??ľ sin đ??ś; f.

â„Žđ?‘Ž = 2đ?‘… sin đ??ľ sin đ??ś. Solution to Problem 55

56. If đ?‘™ is the center of the circle inscribed in triangle đ??´đ??ľđ??ś show that ‖đ??´đ??źâ€– = đ??ľ

đ??ś

4đ?‘… sin 2 sin 2 . Solution to Problem 56

57. Prove the law of sine using the analytic method. Solution to Problem 57

58. Using the law of sine, show that in a triangle the larger side lies opposite to the larger angle. Solution to Problem 58

59. Show that in any triangle đ??´đ??ľđ??ś we have: a.

đ?‘Ž cos đ??ś − đ?‘? cos đ??ľ + cos đ??ś = 0, đ?‘Ž ≠đ?‘?; đ?‘Ž cos đ??ľ − đ?‘? cos đ??´

40

255 Compiled and Solved Problems in Geometry and Trigonometry

sin(đ??´ − đ??ľ) sin đ??ś đ?‘Ž2 − đ?‘? 2 = 2 ; 1 + cos(đ??´ − đ??ľ) cos đ??ś đ?‘Ž + đ?‘? 2 đ??ľ 3đ??ľ đ??ľ đ??ľ c. (đ?‘Ž + đ?‘?) cos + đ?‘Ž cos(đ??´ + ) = 2đ?‘? cos cos . 4 4 2 4

b.

Solution to Problem 59

60. In a triangle đ??´đ??ľđ??ś, đ??´ ∈ 45°, ‖đ??´đ??ľâ€– = đ?‘Ž, ‖đ??´đ??śâ€– =

2√2 3

đ?‘Ž. Show that tan đ??ľ = 2. Solution to Problem 60

61. Let đ??´â€˛, đ??ľâ€˛, đ??śâ€˛ be tangent points of the circle inscribed in a triangle đ??´đ??ľđ??ś with its sides. Show that

đ?œŽ[đ??´â€˛ đ??ľâ€˛ đ??ś ′ ] đ?œŽ[đ??´đ??ľđ??ś]

đ?‘&#x;

= 2đ?‘…. Solution to Problem 61

đ??´

đ?‘Ž

62. Show that in any triangle đ??´đ??ľđ??ś sin 2 ≤ 2√đ?‘?đ?‘? . Solution to Problem 62

63. Solve the triangle đ??´đ??ľđ??ś, knowing its elements đ??´, đ??ľ and area đ?‘†. Solution to Problem 63

4

64. Solve the triangle đ??´đ??ľđ??ś, knowing đ?‘Ž = 13, arc cos 5 , and the corresponding 1

median for side đ?‘Ž, đ?‘šđ?‘Ž = 2 √15√3 . Solution to Problem 64

65. Find the angles of the triangle đ??´đ??ľđ??ś, knowing that đ??ľ − đ??ś =

2đ?œ‹ 3

and đ?‘… = 8đ?‘&#x;,

where đ?‘… and đ?‘&#x; are the radii of the circles circumscribed and inscribed in the triangle. Solution to Problem 65

41

Florentin Smarandache

Solutions Solution to Problem 39.

Let đ?‘‘1 and đ?‘‘2 be the two concurrent lines. We draw 2 parallel lines to đ?‘‘1 located on its both sides at distance đ?‘™. These intersect on đ?‘‘2 at đ??ˇ and đ??ľ, which will be points of the locus to be found, because the sum of the distances đ?‘‘(đ??ľ, đ?‘‘1 ) + đ?‘‘(đ??ľ, đ?‘‘2 ) = đ?‘™ + 0 verifies the condition from the statement. We draw two parallel lines with đ?‘‘2 located at distance đ?‘™ from it, which cut đ?‘‘1 in đ??´ and đ??ś, which are as well points of the locus to be found. The equidistant parallel |đ??ˇđ?‘‚| ≥ |đ?‘‚đ??ľ| lines determine on đ?‘‘2 congruent segments â&#x;š , in the same way đ??´đ??ľđ??śđ??ˇ |đ??´đ?‘‚| ≥ |đ?‘‚đ??ś| is a parallelogram. ∆đ??ľđ?‘‚đ??ś,

‖đ??śđ??śâ€˛â€– = đ?‘‘(đ??ś, đ?‘‘2 ) } â&#x;š ‖đ??śđ??ś ′ ‖ = ‖đ??ľđ??ľâ€˛â€– ‖đ??ľđ??ľâ€˛â€– = đ?‘‘(đ??ľ, đ?‘‘1 ) â&#x;š ∆đ??ľđ?‘‚đ??ś is isosceles.

â&#x;š ||đ?‘‚đ??ś|| = ||đ?‘‚đ??ľ|| â&#x;š đ??´đ??ľđ??śđ??ˇ is a rectangle. Any point đ?‘€ we take on the sides of this rectangle, we have ||đ?‘…1 , đ?‘‘1 || + ||đ?‘€, đ?‘‘2 || = đ?‘™, using the propriety according to which the sum of the distances from a point on the base of an isosceles triangle at the sides is constant and equal to the height that starts from one vertex of the base, namely đ?‘™. Thus the desired locus is rectangle đ??´đ??ľđ??śđ??ˇ.

Solution to Problem 40.

42

255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 41.

Solution to Problem 42.

đ?‘šđ?‘Ž2 = đ?‘? 2 +

đ?‘Ž2 4

đ?‘Ž

− 2 2 đ?‘? cos đ??ľ ; đ?‘Ž2 + đ?‘? 2 − đ?‘? 2 = 4đ?‘? 2 + đ?‘Ž2 − 2đ?‘Ž − 2đ?‘? 2 + 2đ?‘? 2 2đ?‘Žđ?‘? = 2đ?‘? 2 + 2đ?‘? 2 − đ?‘Ž2 = 2(đ?‘? 2 + đ?‘? 2 ) − đ?‘Ž2 .

4đ?‘šđ?‘Ž2 = 4đ?‘? 2 + đ?‘Ž2 − 4đ?‘Žđ?‘? cos đ??ľ = 4đ?‘? 2 + đ?‘Ž2 − 4đ?‘Žđ?‘?

43

Florentin Smarandache

Solution to Problem 43. Using the sine theorem, đ?‘Ž = đ?‘š sin đ??´.

đ??´=đ??ľ+đ??ś đ??´âˆ’đ??ś đ??ľ = − â&#x;š đ??´ − đ??ľ = đ??ś or đ??´ − đ??ś = đ??ľ â&#x;š or â&#x;š 2 2 đ??´+đ??ľ =đ??ś

Solution to Problem 44.

By substitution:

44

2đ??´ = 1800 or â&#x;š 2đ??ś = 1800

đ??´ = 900 or đ??ś = 900

255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 45. a. Using the law of sine,

b.

c.

Therefore,

We solve the system, and find đ??ľ and đ??ś. Then we find đ?‘? =

đ?‘Ž sin đ??ľ sin đ??´

and đ?‘? = đ?‘? − đ?‘‘.

Solution to Problem 46. đ?‘Ž sin đ??´

đ?‘?

đ?‘?

= sin đ??ľ = sin đ??ś = đ?‘š â&#x;š

đ?‘Ž = đ?‘š sin đ??´ ; đ?‘? = đ?‘š sin đ??ľ

đ??´âˆ’đ??ľ đ??´+đ??ľ đ??ś đ?‘Ž − đ?‘? đ?‘š sin đ??´ − đ?‘š sin đ??ľ sin đ??´ − sin đ??ľ 2 sin 2 cos 2 đ??´ − đ??ľ sin 2 = = = = tan đ?‘Ž + đ?‘? đ?‘š sin đ??´ + đ?‘š sin đ??ľ sin đ??´ + sin đ??ľ 2 sin đ??´ + đ??ľ cos đ??´ − đ??ľ 2 cos đ??ś 2 2 2 đ??´âˆ’đ??ľ đ??ś = tan tan . 2 2

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Florentin Smarandache

Solution to Problem 47. Using tangents’ theorem,

So

Solution to Problem 48.

In ∆đ??ľđ??ˇđ??ś we have

46

255 Compiled and Solved Problems in Geometry and Trigonometry

In ∆đ??´đ??ˇđ??ľ,

In ∆đ??´đ??ˇđ??ľ we apply sine’s theorem:

Ě‚ ) and we add it to Or we find đ?œ‡(đ??ˇđ??ľđ??ś

đ?œ‹ 6

.

Solution to Problem 49.

47

Florentin Smarandache

sin đ??´ =

−1+√5 , 4

because đ?‘š(đ??´) < 1800 and sin đ??´ > 0.

Solution to Problem 50.

48

255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 51.

49

Florentin Smarandache

Solution to Problem 52.

At problem 9 we’ve found that

With Heron’s formula, we find the area of each triangle and we add them up.

Solution to Problem 53. The formula for the area of a regular polygon:

50

255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 54.

In ∆đ??ľđ?‘‚đ?‘€:

In ∆đ?‘ đ?‘‚đ??ś:

In ∆đ?‘ƒđ?‘‚đ??ˇ:

Substituting (1), (2), (3) in the given relation:

or

which is impossible.

đ?‘š(complete circle) 2đ?œ‹ = = 7. 2đ?œ‹ Ě‚) đ?‘š(đ??´đ??ľ 7 Thus the polygon has 7 sides. đ?‘›=

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Florentin Smarandache

Solution to Problem 55.

Solution to Problem 56.

52

255 Compiled and Solved Problems in Geometry and Trigonometry

We apply the law of sine in ∆đ??´đ??ľđ??ź:

The law of sine applied in ∆đ??´đ??ľđ??ś:

Solution to Problem 57.

In ∆đ??´đ??śđ??śâ€˛: sin(1800 − đ??´) =

‖đ??śđ??śâ€˛â€– đ?‘?

â&#x;š ‖đ??śđ??ś ′ ‖ = đ?‘? sin đ??´ ; cos(1800 − đ??´) = đ?‘? cos đ??´.

So the coordinates of đ??ś are (−đ?‘? cos đ??´ , đ?‘? sin đ??´). The center of the inscribed circle is at the intersection of the perpendicular lines drawn through the midpoints of sides đ??´đ??ľ and đ??´đ??ś.

The equation of the line đ??¸đ?‘‚:

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Florentin Smarandache

If we redo the calculus for the same draw, we have the following result: (đ?‘? cos đ??´ , đ?‘? sin đ??´).

using the law of cosine.

Solution to Problem 58. đ?‘Ž đ?‘? đ?‘? = = = 2đ?‘…. sin đ??´ sin đ??ľ sin đ??ś We suppose that đ?‘Ž > đ?‘?. Let’s prove that đ??´ > đ??ľ. đ?‘Ž đ?‘? đ?‘Ž sin đ??´ = ⇒ = sin đ??´ sin đ??ľ đ?‘? sin đ??ľ} â&#x;š sin đ??´ > 1 â&#x;š đ??´, đ??ľ, đ??ś ∈ (0, đ?œ‹) â&#x;š sin đ??ľ > 0 â&#x;š sin đ??´ > sin đ??ľ đ?‘Ž sin đ??ľ đ?‘Ž>đ?‘?⇒ >1 đ?‘? đ??´âˆ’đ??ľ đ??´+đ??ľ đ??´ + đ??ľ 1800 − đ??ś â&#x;š sin đ??´ − sin đ??ľ > 0 â&#x;š 2 sin cos >0â&#x;š = 2 2 2 2 đ??ś = 900 − . 2 54

255 Compiled and Solved Problems in Geometry and Trigonometry

cos (−

đ??´+đ??ľ 2

đ??ś

đ??ś

= cos (90circ − 2 ) = sin 2 > 0, therefore

đ?œ‹ đ??´âˆ’đ??ľ đ?œ‹ < < ). 2 2 2

Solution to Problem 59.

b. We transform the product into a sum:

From (1) and (2) â&#x;š

We consider the last two terms:

55

đ??´âˆ’đ??ľ 2

> 0 â&#x;š đ??´ > đ??ľ;

Florentin Smarandache

Solution to Problem 60. We apply the law of cosines in triangle ABC:

Solution to Problem 61. ‖đ??źđ??´â€– = ‖đ??źđ??ľâ€– = ‖đ??źđ??śâ€– = đ?‘&#x; đ??źđ??śâ€˛ ⊼ đ??´đ??ľ} ⇒ đ??źđ??´â€˛đ??ľâ€˛đ??śâ€˛ inscribable quadrilateral đ??źđ??´â€˛ ⊼ đ??ľđ??ś Ě‚ ) = sin đ??ľĚ‚ đ?‘š(đ??´â€˛đ??źđ??śâ€˛) = 180 − đ??ľĚ‚ â&#x;š sin(đ??´â€˛đ??źđ??śâ€˛ Ě‚ = sin đ??ś and đ??śâ€˛đ??źđ??ľâ€˛ Ě‚ = sin đ??´. Similarly, đ??´â€˛đ??źđ??ľâ€˛ 56

255 Compiled and Solved Problems in Geometry and Trigonometry

In the same way,

Solution to Problem 62. đ??´

0 < đ??´ ⇒< 2 <

đ?œ‹ 2

đ??´

⇒ sin 2 > 0 ;

đ??´ 1 − cos đ??´ sin = √ â&#x;š 2 2

đ??´ 1− â&#x;š sin2 = 2

đ??´ 1 − cos đ??´ = 2 2 } đ?‘? 2 + đ?‘? 2 − đ?‘Ž2 cos đ??´ = 2đ?‘?đ?‘? sin2

đ?‘? 2 − đ?‘Ž2 + đ?‘? 2 đ?‘Ž2 − (đ?‘? − đ?‘?)2 đ?‘Ž2 đ??´ đ?‘Ž 2đ?‘?đ?‘? = ≤ â&#x;š sin ≤ . 2 4đ?‘?đ?‘? 4đ?‘?đ?‘? 2 2√đ?‘?đ?‘?

Solution to Problem 63. đ??ś = đ?œ‹ − (đ??´ + đ??ľ) đ?‘†=

đ?‘Ž 2 sin đ??ľ sin đ??ś 2 sin đ??´

} â&#x;š We find đ?‘Ž. đ??´, đ??ľ, đ??ś are known đ?‘Ž sin đ?‘Ž

đ?‘?

= sin đ?‘? â&#x;š đ?‘? =

đ?‘Ž sin đ??ľ . sin đ??´

In the same way, we find đ?‘?.

Solution to Problem 64.

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Florentin Smarandache

đ?‘? 2 + đ?‘? 2 = 841} â&#x;š {đ?‘? = 21 or {đ?‘? = 20 đ?‘? = 20 đ?‘? = 21 đ?‘?đ?‘? = 420

We find đ??ľ.

We find the sum. Or

Solution to Problem 65.

We already know that đ?‘&#x; đ??´ đ??ľ đ??ś = 4 sin sin sin đ?‘… 2 2 2

đ??´

We write sin 2 = đ?‘Ą. We have

58

255 Compiled and Solved Problems in Geometry and Trigonometry

From this system we find đ??ľ and đ??ś.

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Florentin Smarandache

Other Problems in Geometry and Trigonometry (10 grade) th

66. Show that a convex polygon can’t have more than three acute angles. Solution to Problem 66

67. Let đ??´đ??ľđ??ś be a triangle. Find the locus of points đ?‘€ ∈ (đ??´đ??ľđ??ś), for which đ?œŽ[đ??´đ??ľđ?‘€] = đ?œŽ[đ??´đ??śđ?‘€]. Solution to Problem 67

68. A convex quadrilateral đ??´đ??ľđ??śđ??ˇ is given. Find the locus of points đ?‘€ ∈ đ?‘–đ?‘›đ?‘Ą. đ??´đ??ľđ??śđ??ˇ, for which đ?œŽ[đ?‘€đ??ľđ??śđ??ˇ] = đ?œŽ[đ?‘€đ??ľđ??´đ??ˇ]. Solution to Problem 68

69. Determine a line đ?‘€đ?‘ , parallel to the bases of a trapezoid đ??´đ??ľđ??śđ??ˇ (đ?‘€ ∈ |đ??´đ??ˇ|, đ?‘ ∈ |đ??ľđ??ś|) such that the difference of the areas of [đ??´đ??ľđ?‘ đ?‘€] and [đ?‘€đ?‘ đ??śđ??ˇ] to be equal to a given number. Solution to Problem 69

70. On the sides of ∆đ??´đ??ľđ??ś we take the points đ??ˇ, đ??¸, đ??š such that

đ??ľđ??ˇ đ??ˇđ??ś

đ??śđ??¸

đ??´đ??š

= đ??¸đ??´ = đ??šđ??ľ = 2.

Find the ratio of the areas of triangles đ??ˇđ??¸đ??š and đ??´đ??ľđ??ś. Solution to Problem 70

đ?‘Ž

71. Consider the equilateral triangle đ??´đ??ľđ??ś and the disk [đ??ś (đ?‘‚, 3)], where đ?‘‚ is the orthocenter of the triangle and đ?‘Ž = ‖đ??´đ??ľâ€–. Determine the area [đ??´đ??ľđ??ś] − đ?‘Ž

[đ??ś (đ?‘‚, 3)]. Solution to Problem 71

60

255 Compiled and Solved Problems in Geometry and Trigonometry

72. Show that in any triangle đ??´đ??ľđ??ś we have: a. 1 + cos đ??´ cos(đ??ľ − đ??ś) =

đ?‘? 2 +đ?‘? 2 4đ?‘… 2

;

b. (đ?‘? 2 + đ?‘? 2 = đ?‘Ž2 ) tan đ??´ = 4đ?‘†; c.

đ?‘?+đ?‘?

đ??´

sin( +đ??ś)

2 ; đ??´ = sin(đ??´+đ??ľ)

2đ?‘? cos

2

đ??´

đ??ľ

đ??ľ

đ??ś

d. đ?‘? = đ?‘&#x; (cot 2 + cot 2 + cot đ??ś2); đ??´

e. cot 2 + cot 2 + cot 2 =

đ?‘? đ?‘&#x;

. Solution to Problem 72

73. If đ??ť is the orthocenter of triangle đ??´đ??ľđ??ś, show that: a. ‖đ??´đ??ťâ€– = 2đ?‘… cos đ??´; b. đ?‘Žâ€–đ??´đ??ťâ€– + đ?‘?‖đ??ľđ??ťâ€– + đ?‘?‖đ??śđ??ťâ€– = 4đ?‘†. Solution to Problem 73

74. If đ?‘‚ is the orthocenter of the circumscribed circle of triangle đ??´đ??ľđ??ś and đ??ź is the center of the inscribed circle, show that ‖đ?‘‚đ??źâ€–2 = đ?‘…(đ?‘… – 2đ?‘&#x;). Solution to Problem 74

75. Show that in any triangle đ??´đ??ľđ??ś we have: cos 2

đ??ľâˆ’đ??ś 2

≼

2đ?‘&#x; đ?‘…

. Solution to Problem 75

1

1

76. Find � � + � � knowing that � + � = 2 sin �. Solution to Problem 76

77. Solve the equation: (đ?‘§ + 1)đ?‘› − (đ?‘§ − 1)đ?‘› = 0. Solution to Problem 77

1

3

78. Prove that if � < 2 then |(1 + �)� 3 + ��| ≤ 4 . Solution to Problem 78

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Florentin Smarandache

79. One gives the lines � and �′. Show that through each point in the space passes a perpendicular line to � and �′. Solution to Problem 79

80. There are given the lines đ?‘‘ and đ?‘‘′, which are not in the same plane, and the points đ??´ ∈ đ?‘‘, đ??ľ ∈ đ?‘‘′. Find the locus of points đ?‘€ for which prđ?‘‘ đ?‘€ = đ??´ and prđ?‘‘′ đ?‘€ = đ??ľ. Solution to Problem 80

Ě‚ equally distant 81. Find the locus of the points inside a trihedral angle đ?‘Žđ?‘?đ?‘? from the edges of đ?‘Ž, đ?‘?, đ?‘?. Solution to Problem 81

82. Construct a line which intersects two given lines and which is perpendicular to another given line. Solution to Problem 82

83. One gives the points đ??´ and đ??ľ located on the same side of a plane; find in this plane the point for which the sum of its distances to đ??´ and đ??ľ is minimal. Solution to Problem 83

84. Through a line draw a plane onto which the projections of two lines to be parallel. Solution to Problem 84

85. Consider a tetrahedron [đ??´đ??ľđ??śđ??ˇ] and centroids đ??ż, đ?‘€, đ?‘ of triangles đ??ľđ??śđ??ˇ, đ??śđ??´đ??ˇ, đ??´đ??ľđ??ˇ. a. Show that (đ??´đ??ľđ??ś) âˆĽ (đ??żđ?‘€đ?‘ ); b. Find the ratio

đ?œŽ[đ??´đ??ľđ??ś] đ?œŽ[đ??żđ?‘€đ?‘ ]

. Solution to Problem 85

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255 Compiled and Solved Problems in Geometry and Trigonometry

86. Consider a cube [đ??´đ??ľđ??śđ??ˇđ??´â€˛đ??ľâ€˛đ??śâ€˛đ??ˇâ€˛]. The point đ??´ is projected onto đ??´â€˛đ??ľ, đ??´â€˛đ??ś, đ??´â€˛đ??ˇ respectively in đ??´1 , đ??´2 , đ??´3 . Show that: a. đ??´â€˛ đ??ś ⊼ (đ??´1 đ??´2 đ??´3 ); b. đ??´đ??´1 ⊼ đ??´1 đ??´2 , đ??´đ??´3 ⊼ đ??´3 đ??´2 ; c. đ??´đ??´1 đ??´2 đ??´3 is an inscribable quadrilateral. Solution to Problem 86

Ě‚ = 900 ) Ě‚ )) = đ?‘š((đ??´đ??ľđ??ˇ) 87. Consider the right triangles đ??ľđ??´đ??ś and đ??´đ??ľđ??ˇ (đ?‘š(đ??ľđ??´đ??ś located on perpendicular planes đ?‘€ and đ?‘ , being midpoints of segments [đ??´đ??ľ], [đ??śđ??ˇ]. Show that đ?‘€đ?‘ ⊼ đ??śđ??ˇ. Solution to Problem 87

88. Prove that the bisector half-plane of a dihedral angle inside a tetrahedron divides the opposite edge in proportional segments with the areas of the adjacent faces. Solution to Problem 88

89. Let đ??´ be a vertex of a regular tetrahedron and đ?‘ƒ, đ?‘„ two points on its Ě‚ ) ≤ 600 . surface. Show that đ?‘š(đ?‘ƒđ??´đ?‘„ Solution to Problem 89

90. Show that the sum of the measures of the dihedral angles of a tetrahedron is bigger than 360°. Solution to Problem 90

91. Consider lines đ?‘‘1 , đ?‘‘2 contained in a plane đ?›ź and a line đ??´đ??ľ which intersects plane đ?›ź at point đ??ś. A variable line, included in đ?›ź and passing through đ??ś all đ?‘‘1 , đ?‘‘2 respectively at đ?‘€đ?‘ . Find the locus of the intersection đ??´đ?‘€ ∊ đ??ľđ?‘ . In which case is the locus an empty set? Solution to Problem 91

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Florentin Smarandache

92. A plane đ?›ź intersects sides [đ??´đ??ľ], [đ??ľđ??ś], [đ??śđ??ˇ], [đ??ˇđ??´] of a tetrahedron [đ??´đ??ľđ??śđ??ˇ] at points đ??ż, đ?‘€, đ?‘ , đ?‘ƒ. Prove that ‖đ??´đ??żâ€– ∙ ‖đ??ľđ?‘€â€– ∙ ‖đ??śđ?‘ ‖ ∙ ‖đ?‘ƒđ??ˇâ€– = ‖đ??ľđ??żâ€– ∙ ‖đ??śđ?‘€â€– ∙ ‖đ??ˇđ?‘ ‖ ∙ ‖đ??´đ?‘ƒâ€–. Solution to Problem 92

93. From a point đ??´ located outside a plane đ?›ź, we draw the perpendicular line đ??´đ?‘‚, đ?‘‚ ∈ đ?›ź, and we take đ??ľ, đ??ś ∈ đ?›ź. Let đ??ť, đ??ť1 be the orthocenters of triangles đ??´đ??ľđ??ś, đ?‘‚đ??ľđ??ś; đ??´đ??ˇ and đ??ľđ??¸ heights in triangle đ??´đ??ľđ??ś; and đ??ľđ??¸1 height in triangle đ?‘‚đ??ľđ??ś. Show that: a. đ??ťđ??ť1 ⊼ (đ??´đ??ľđ??ś); đ?‘‚đ??´

đ??ˇđ??ť1

đ??´đ??ˇ

đ??ť1 đ??ľ

b. ‖ ‖ ∙ ‖

‖∙‖

đ??ľđ??¸

đ??¸đ??¸1

‖ = 1. Solution to Problem 93

94. Being given a tetrahedron [đ??´đ??ľđ??śđ??ˇ] where đ??´đ??ľ ⊼ đ??śđ??ˇ and đ??´đ??ś ⊼ đ??ľđ??ˇ, show that: a. ‖đ??´đ??ľâ€–2 + ‖đ??śđ??ˇâ€–2 = ‖đ??ľđ??śâ€–2 + ‖đ??´đ??ˇâ€–2 = ‖đ??śđ??´â€–2 + ‖đ??ľđ??ˇâ€–2 ; b. The midpoints of the 6 edges are located on a sphere. Solution to Problem 94

95. It is given a triangular prism [đ??´đ??ľđ??śđ??´â€˛đ??ľâ€˛đ??śâ€˛] which has square lateral faces. Let đ?‘€ be a mobile point [đ??´đ??ľâ€˛], đ?‘ the projection of đ?‘€ onto (đ??ľđ??śđ??śâ€˛) and đ??´Ęş the midpoint of [đ??ľâ€˛đ??śĘş]. Show that đ??´â€˛đ?‘ and đ?‘€đ??´Ęş intersect in a point đ?‘ƒ and find the locus of đ?‘ƒ. Solution to Problem 95

96. We have the tetrahedron [đ??´đ??ľđ??śđ??ˇ] and let đ??ş be the centroid of triangle đ??ľđ??śđ??ˇ. Show that if đ?‘€ ∈ đ??´đ??ş then đ?œ?[đ?‘€đ??şđ??ľđ??ś] = đ?œ?[đ?‘€đ??şđ??śđ??ˇ] = đ?œ?[đ?‘€đ??şđ??ˇđ??ľ]. Solution to Problem 96

97. Consider point đ?‘€ ∈ the interior of a trirectangular tetrahedron with its vertex in đ?‘‚. Draw through đ?‘€ a plane which intersects the edges of the

64

255 Compiled and Solved Problems in Geometry and Trigonometry

respective tetrahedron in points đ??´, đ??ľ, đ??ś so that đ?‘€ is the orthocenter of ∆đ??´đ??ľđ??ś. Solution to Problem 97

98. A pile of sand has as bases two rectangles located in parallel planes and trapezoid side faces. Find the volume of the pile, knowing the dimensions đ?‘Žâ€˛, đ?‘?′ of the small base, đ?‘Ž, đ?‘? of the larger base, and â„Ž the distance between the two bases. Solution to Problem 98

99. A pyramid frustum is given, with its height â„Ž and the areas of the bases đ??ľ and đ?‘?. Unite any point đ?œŽ of the larger base with the vertices đ??´, đ??ľ, đ??´â€˛, đ??ľâ€˛ of a side face. Show that đ?œ?[đ?‘‚đ??´â€˛đ??ľâ€˛đ??´] =

√6 √đ??ľ

đ?œ?[đ?‘‚đ??´đ??ľđ??ľâ€˛]. Solution to Problem 99

100. A triangular prism is circumscribed to a circle of radius đ?‘…. Find the area and the volume of the prism. Solution to Problem 100

101. A right triangle, with its legs đ?‘? and đ?‘? and the hypotenuse đ?‘Ž, revolves by turns around the hypotenuse and the two legs, đ?‘‰1 , đ?‘‰2 , đ?‘‰3 ; đ?‘†1 , đ?‘†2 , đ?‘†3 being the volumes, respectively the lateral areas of the three formed shapes, show that: a. b.

1 �12 �2 �3

1

1

= �2 = �2; 2

�3

+� = 2

3

�2 +�3 �1

. Solution to Problem 101

102. A factory chimney has the shape of a cone frustum and 10m height, the bases of the cone frustum have external lengths of 3,14m and 1,57m, and the wall is 18cm thick. Calculate the volume of the chimney. Solution to Problem 102

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Florentin Smarandache

103. A regular pyramid, with its base a square and the angle from the peak of a side face of measure � is inscribed in a sphere of radius �. Find: a. the volume of the inscribed pyramid; b. the lateral and total area of the pyramid; c. the value � when the height of the pyramid is equal to the radius of the sphere. Solution to Problem 103

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255 Compiled and Solved Problems in Geometry and Trigonometry

Solutions Solution to Problem 66.

Let đ??´1 , đ??´2 ‌ đ??´đ?‘› the vertices of the convex polygon. Let’s assume that it has four acute angles. The vertices of these angles form a convex quadrilateral đ??´đ?‘™ đ??´đ?‘˜ đ??´đ?‘š đ??´đ?‘› . Due to the fact that the polygon is convex, the segments |đ??´đ?‘™ đ??´đ?‘˜ |, |đ??´đ?‘˜ đ??´đ?‘š |, |đ??´đ?‘š đ??´đ?‘› |, |đ??´đ?‘› đ??´đ?‘™ | are inside the initial polygon. We find that the angles of the quadrilateral are acute, which is absurd, because their sum is 360°.

Another solution: We assume that đ??´đ?‘™ đ??´đ?‘˜ đ??´đ?‘š đ??´đ?‘› is a convex polygon with all its angles acute â&#x;š the sum of the external angles is bigger than 360°, which is absurd (the sum of the measures of the external angles of a convex polygon is 360°).

Solution to Problem 67.

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Florentin Smarandache

Let |đ??´đ??´â€˛ | be the median from đ??´ and đ??śđ?‘„ ⊼ đ??´đ??´â€˛ , đ??ľđ?‘ƒ ⊼ đ??´đ??´â€˛. ∆đ??ľđ??´â€˛đ?‘ƒ ≥ đ??śđ??´â€˛đ?‘„ because: Ě‚ ≥ đ??ľđ??śđ?‘„ Ě‚ alternate interior đ?‘ƒđ??ľđ??ś ′đ?‘„ Ě‚ ≥ đ??śđ??´ {đ?‘ƒđ??´â€˛đ??ľ Ě‚ vertical angles đ??ľđ??´â€˛ ≥ đ??´â€˛ đ??ś â&#x;š ||đ??ľđ?‘ƒ|| = ||đ?‘„đ??ś|| and by its construction đ??ľđ?‘ƒ ⊼ đ??´đ??´â€˛, đ??śđ?‘„ ⊼ đ??´đ??´â€˛. The desired locus is median |đ??´đ??´â€˛|. Indeed, for any đ?‘€ ∈ |đ??´đ??´â€˛| we have đ?œŽ[đ??´đ??ľđ?‘€] = đ?œŽ[đ??´đ??śđ?‘€], because triangles đ??´đ??ľđ?‘€ and đ??´đ??śđ?‘€ have a common side |đ??´đ?‘€| and its corresponding height equal ||đ??ľđ?‘ƒ|| = ||đ?‘„đ??ś||.

Vice-versa. If đ?œŽ[đ??´đ??ľđ?‘€] = đ?œŽ[đ??´đ??śâ€˛đ?‘€], let’s prove that đ?‘€ ∈ |đ??´đ??´â€˛|. Indeed: đ?œŽ[đ??´đ??ľđ?‘€] đ?œŽ[đ??´đ??śđ?‘€] ⇒ đ?‘‘ (đ??ľ, đ??´đ?‘€) = đ?‘‘(đ??ś, đ??´đ?‘€), because |đ??´đ?‘€| is a common side, đ?‘‘ (đ??ľ, đ??´đ?‘€) = ||đ??ľđ?‘ƒ|| and đ?‘‘(đ??ś, đ??´đ?‘€) = ||đ??śđ?‘„|| and both are perpendicular to đ??´đ?‘€ â&#x;š đ?‘ƒđ??ľđ?‘„đ??ś is a parallelogram, the points đ?‘ƒ, đ?‘€, đ?‘„ are collinear (đ?‘ƒ, đ?‘„ the feet of the perpendicular lines from đ??ľ and đ??ś to đ??´đ?‘€). In parallelogram đ?‘ƒđ??ľđ?‘„đ??ś we have |đ?‘ƒđ?‘„| and |đ??ľđ??ś| diagonals â&#x;š đ??´đ?‘€ passes through the middle of |đ??ľđ??ś|, so đ?‘€ ∈ |đ??´đ??´â€˛|, the median from đ??´.

Solution to Problem 68.

Let đ?‘‚ be the midpoint of diagonal |đ??´đ??ś| â&#x;š ‖đ??´đ?‘‚‖ = ‖đ?‘‚đ??śâ€–. đ?œŽ[đ??´đ?‘‚đ??ˇ] = đ?œŽ[đ??śđ?‘‚đ??ˇ] (1) ‖đ??´đ?‘‚‖ = ‖đ?‘‚đ??śâ€– Because { ‖đ?‘‚đ??ˇâ€˛â€– common height đ?œŽ[đ??´đ?‘‚đ??ľ] = đ?œŽ[đ??śđ?‘‚đ??ľ] (2) 68

255 Compiled and Solved Problems in Geometry and Trigonometry

the same reasons; we add up (1) and (2) â&#x;š đ?œŽ[đ??´đ??ˇđ?‘‚đ??ľ] = đ?œŽ[đ??ˇđ??śđ??ľđ?‘‚] (3), so đ?‘‚ is a point of the desired locus. We construct through đ?‘‚ a parallel to đ??ľđ??ˇ until it cuts sides |đ??ľđ??ś| and |đ??ˇđ??ś| at đ?‘ƒ respectively đ?‘„. The desired locus is |đ?‘ƒđ?‘„|. Indeed (∀)đ?‘€ ∈ |đ?‘ƒđ?‘„| we have:

đ?œŽ[đ??ľđ??ˇđ?‘‚] = đ?œŽ[đ??ľđ??ˇđ?‘€] because đ?‘€ and đ?‘„ belongs to a parallel to đ??ľđ??ˇ.

đ??ľ, đ??ˇ ∈ a parallel to đ?‘‚đ?‘€. đ?œŽ[đ?‘€đ??ľđ??´đ??ˇ] = đ?œŽ[đ??´đ??ľđ?‘‚đ??ˇ] So and đ?œŽ[đ??ľđ??śđ??ˇđ?‘€] = đ?œŽ[đ?‘‚đ??ľđ??śđ??ˇ]} â&#x;š đ?œŽ[đ?‘€đ??ľđ??´đ??ˇ] = đ?œŽ[đ??ľđ??śđ??ˇđ?‘€]. and from (3)

Vice-versa: If đ?œŽ[đ?‘€đ??ľđ??śđ??ˇ] = đ?œŽ[đ?‘€đ??ľđ??´đ??ˇ], let’s prove that đ?‘€ ∈ parallel line through đ?‘‚ to đ??ľđ??ˇ. Indeed: đ?œŽ[đ??ľđ??śđ??ˇđ?‘€] = đ?œŽ[đ?‘€đ??ľđ??´đ??ˇ] đ?œŽ[đ??´đ??ľđ??śđ??ˇ] (2). } â&#x;š đ?œŽ[đ?‘€đ??ľđ??śđ??ˇ] = đ?œŽ[đ?‘€đ??ľđ??´đ??ˇ] = 2 and because đ?œŽ[đ??ľđ??śđ??ˇđ?‘€] + đ?œŽ[đ?‘€đ??ľđ??´đ??ˇ] = đ?œŽ[đ??´đ??ľđ??śđ??ˇ] So, from (1) and (2) â&#x;š đ?œŽ[đ?‘€đ??ľđ??´đ??ˇ] = đ?œŽ[đ??´đ??ľđ?‘‚đ??ˇ] â&#x;š đ?œŽ[đ??´đ??ľđ??ˇ] + đ?œŽ[đ??ľđ??ˇđ?‘‚] = đ?œŽ[đ??´đ??ľđ??ˇ] + đ?œŽ[đ??ľđ??ˇđ?‘€] â&#x;š đ?œŽ[đ??ľđ??ˇđ?‘€] â&#x;š đ?‘€ and đ?‘‚ are on a parallel to đ??ľđ??ˇ.

Solution to Problem 69.

We write ||đ??¸đ??´|| = đ?‘Ž and ||đ??¸đ??ˇ|| = đ?‘?, ||đ??¸đ?‘€|| = đ?‘Ľ.

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Florentin Smarandache

We subtract (2) from (3)

We subtract (2) from (4)

from the hypothesis

đ?‘†đ?‘?2

From the relation (3), by writing [đ??´đ??ľđ??śđ??ˇ] − đ?‘† â&#x;š đ?œŽ[đ??¸đ??śđ??ˇ] = đ?‘Ž2 −đ?‘?2 . We substitute this in the relation of đ?‘ĽÂ˛ and we obtain:

and taking into consideration that ||đ??¸đ?‘€|| = ||đ??ˇđ?‘€|| + đ?‘?, we have

so we have the position of point đ?‘€ on the segment |đ??ˇđ??´| (but it was sufficient to find the distance ||đ??¸đ?‘€||).

Solution to Problem 70.

We remark from its construction that đ??¸đ?‘„||đ??´đ??ľ||đ?‘…đ??ˇ, more than that, they are equidistant parallel lines. Similarly, đ??¸đ?‘„, đ?‘ƒđ??ˇ, đ??´đ??ś and đ??´đ??ľ, đ??¸đ?‘„, đ?‘…đ??ˇ are also equidistant parallel lines.

70

255 Compiled and Solved Problems in Geometry and Trigonometry

We write đ?œŽ[đ??ľđ??šđ?‘„] = đ?‘†. Based on the following properties: 

two triangles have equal areas if they have equal bases and the same height;



two triangles have equal areas if they have the same base and the third peak on a parallel line to the base,

we have:

by addition

⇒ So

đ?œŽ[đ??ˇđ??¸đ??š] đ?œŽ[đ??´đ??ľđ??ś]

=

3� 9�

=

1 3

.

(If necessary the areas � can be arranged).

Solution to Problem 71.

‖đ?‘‚đ??ľâ€– =

đ?‘Žâˆš3 6

(đ??ľđ??ľâ€˛ median)

In ∆đ?‘€đ?‘‚đ??ľâ€˛:

Ě‚)=đ?œ‹ . So (đ?‘€đ?‘‚đ?‘ 3 We mark with đ?›´ the disk surface bordered by a side of the triangle outside the triangle.

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Florentin Smarandache

đ?œŽ[đ?›´] = đ?œŽ[circle sector đ?‘€đ?‘‚đ?‘ ] − đ?œŽ[đ?‘€đ?‘‚đ?‘ ] đ?œ‹đ?‘Ž2 đ?‘Ž2 đ?œ‹đ?‘Ž2 đ?‘Ž2 √3 đ?‘Ž2 đ?œ‹ √3 = − sin 600 = − = ∙ ( − ). 9∙6 9∙2 9∙6 4∙9 18 3 2 If through the disk area we subtract three times đ?œŽ[đ?›´], we will find the area of the disk fraction from the interior of đ??´đ??ľđ??ś. So the area of the disk surface inside đ??´đ??ľđ??ś is:

The desired area is obtained by subtracting the calculated area form đ?œŽ[đ??´đ??ľđ??ś]. So:

Solution to Problem 72. a. 1 + cos đ??´ ∙ cos(đ??ľ − đ??ś) =

đ?‘? 2 +đ?‘? 2 4đ?‘… 2

4�

b. We prove that tan đ??´ = đ?‘?2 +đ?‘? 2−đ?‘Ž2 .

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255 Compiled and Solved Problems in Geometry and Trigonometry

â&#x;š

All terms reduce.

đ??´

đ??ľ

đ?‘?

2

2

đ?‘&#x;

d. Let’s prove that cot + cot + cot đ??ś2 =

Indeed

We now have to prove that:

Solution to Problem 73.

73

.

Florentin Smarandache

a. In triangle đ??´đ??ľđ??ľâ€˛: ‖đ??´đ??ľâ€˛â€– = đ?‘? cos đ??´ In triangle đ??´đ??ťđ??ľâ€˛:

We used:

Solution to Problem 74.

Using the power of point đ??ź in relation to circle đ??ś(đ?‘‚, đ?‘…)

Taking into consideration (1), we have ‖đ??źđ??´â€– ∙ ‖đ??źđ??ˇâ€– = đ?‘… 2 − ‖đ?‘‚đ??źâ€–². We now find the distances ||IA|| and ||ID|| In triangle ∆đ??źđ??´đ?‘ƒ,

Ě‚ ) = đ?œ‡(đ??ˇđ??ľđ??ź Ě‚ ) have the same measure, more exactly: We also find ‖đ??źđ??ˇâ€–: đ?œ‡(đ??ľđ??źđ??ˇ

74

255 Compiled and Solved Problems in Geometry and Trigonometry

In ∆đ??´đ??ľđ??ˇ according to the law of sine, we have:

So taking into consideration (3),

Returning to the relation ‖đ??źđ??´â€– ∙ ‖đ??źđ??ˇâ€– = đ?‘… 2 − ‖đ?‘‚đ??źâ€–². with (2) and (4) we have:

Solution to Problem 75.

Note. We will have to show that

Indeed:

(by Heron’s formula).

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Florentin Smarandache

Solution to Problem 76.

So:

We calculate for �1 and �2 :

so � � +

1 ��

takes the same value for �1 and for �2 and it is enough if we

calculate it for �1 .

Analogously:

Solution to Problem 77.

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255 Compiled and Solved Problems in Geometry and Trigonometry

(we substitute −1 with đ?‘–² at denominator)

Solution to Problem 78.

Solution to Problem 79. We construct đ?›ź ⊼ đ?‘‘ and đ??´ ∈ đ?›ź . The so constructed plane is unique. Similarly we construct đ?›˝ ⊼ đ?‘‘′ and đ??´ ⊼ đ?›˝, đ?›ź ∊ đ?›˝ = đ?‘Ž ∋ đ??´.

From

đ?›źâŠĽđ?‘‘⇒đ?‘‘⊼đ?‘Ž } â&#x;š đ?‘Ž is a line which passes through đ??´ and is perpendicular đ?›˝ ⊼ đ?‘‘′ ⇒ đ?‘‘′ ⊼ đ?‘Ž

to � and �′. The line � is unique, because � and � constructed as above are unique.

Solution to Problem 80. We construct plane đ?›ź such that đ??´ ∈ đ?›ź and đ?‘‘ ⊼ đ?›ź. We construct plane đ?›˝ such that đ??ľ ∈ đ?›˝ and đ?‘‘′ ⊼ đ?›˝.

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Florentin Smarandache

The so constructed planes đ?›ź and đ?›˝ are unique. Let đ?‘Ž = đ?›ź ∊ đ?›˝ â&#x;š đ?‘Ž ⊂ đ?›ź so (∀) đ?‘€ ∈ đ?‘Ž has the property prđ?‘‘ đ?‘€ = đ??´. đ?›ź ⊂ đ?›˝ â&#x;š (∀) đ?‘€ ∈ đ?‘Ž has the property đ?‘?đ?‘&#x;đ?‘‘′ đ?‘€ = đ??ľ.

Vice-versa. If there is a point đ?‘€ in space such that prđ?‘‘ đ?‘€ = đ??´ and đ?‘?đ?‘&#x;đ?‘‘′ đ?‘€ = đ??ľ â&#x;š đ?‘€ ∈ đ?‘Ž and đ?‘€ ∈ đ?›˝ â&#x;š đ?‘€ ∈ đ?›ź ∊ đ?›˝ â&#x;š đ?‘€ ∈ đ?‘Ž (đ?›ź and đ?›˝ previously constructed).

Solution to Problem 81.

Let đ??´ ∈ đ?‘Ž, đ??ľ ∈ đ?‘?, đ??ś ∈ đ?‘? such that ‖đ?‘‚đ??´â€– = ‖đ?‘‚đ??ľâ€– = ‖đ?‘‚đ??śâ€–. Triangles đ?‘‚đ??´đ??ľ, đ?‘‚đ??ľđ??ś, đ?‘‚đ??´đ??ś are isosceles. The mediator planes of segments ‖đ??´đ??ľâ€–, ‖đ??´đ??śâ€–, ‖đ??ľđ??ś ‖ pass through đ?‘‚ and đ?‘‚′ (the center of the circumscribed circle of triangle đ??´đ??ľđ??ś). Ray |đ?‘‚đ?‘‚′| is the desired locus. Indeed (∀) đ?‘€ ∈ |đ?‘‚đ?‘‚′| â&#x;š đ?‘€ ∈ mediator plane of segments |đ??´đ??ľ|, |đ??´đ??ś| and |đ??ľđ??ś| â&#x;š đ?‘€ is equally distant from đ?‘Ž, đ?‘? and đ?‘?.

Vice-versa: (∀) đ?‘€ with the property: đ?‘‘(đ?‘€, đ?‘Ž) = đ?‘‘(đ?‘€, đ?‘?) = đ?‘‘(đ?‘€, đ?‘?) â&#x;š đ?‘€ ∈ mediator plan, mediator planes of segments |đ??´đ??ľ|, |đ??´đ??ś| and |đ??ľđ??ś| â&#x;š đ?‘€ ∈ the intersection of these planes â&#x;š đ?‘€ ∈ |đ?‘‚đ?‘‚′|. 78

255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 82.

Let đ?‘Ž, đ?‘?, đ?‘? be the 3 lines in space. I.

We assume đ?‘Ž ⊼ đ?‘? and đ?‘? ⊼ đ?‘?. Let đ?›ź be a plane such that:

The construction is possible because ⊼ đ?‘? and đ?‘? ⊼ đ?‘?. Line đ??´đ??ľ meets đ?‘Ž on đ?‘? and it is perpendicular to đ?‘?, because đ??´đ??ľ ⊂ đ?›ź and đ?‘? ⊼ đ?›ź. II.

If đ?‘Ž ⊼ đ?‘? or đ?‘? ⊼ đ?‘?, the construction is not always possible, only if plane đ?‘?(đ?‘Ž, đ?‘?) is perpendicular to đ?‘?.

III.

If đ?‘Ž ⊼ đ?‘? and đ?‘? ⊼ đ?‘?, we construct plane đ?‘Ž ⊼ đ?‘? so that đ?‘Ž ⊂ đ?›ź and đ?‘? ⊂ đ?›ź ≠∅. Any point on line a connected with point đ?‘? ∊ đ?›ź is a desired line.

Solution to Problem 83. We construct đ??´â€˛ the symmetrical point of đ??´ in relation to đ?›ź . đ??´â€˛ and đ??ľ are on different half-spaces, đ?›ź ∊ |đ??´â€˛ đ??ľ| = đ?‘‚.

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Florentin Smarandache

đ?‘‚ is the desired point, because ||đ?‘‚đ??´|| + ||đ?‘‚đ??ľ|| = ||đ?‘‚đ??´â€˛|| + ||đ?‘‚đ??ľ|| is minimal when đ?‘‚ ∈ |đ??´â€˛đ??ľ|, thus the desired point is đ?‘‚ = |đ??´â€˛đ??ľ| ∊ đ?›ź.

Solution to Problem 84.

Let đ?‘Ž, đ?‘?, đ?‘‘ be the 3 given lines and through đ?‘‘ we construct a plane in which đ?‘Ž and đ?‘? to be projected after parallel lines. Let đ??´ be an arbitrary point on đ?‘Ž. Through đ??´ we construct line đ?‘?′||đ?‘?. It results from the figure đ?‘?||đ?›ź, đ?›ź = đ?‘?(đ?‘Ž, đ?‘?′). Let đ?›˝ such that đ?‘‘ ⊂ đ?›˝ and đ?›˝ ⊼ đ?›ź. Lines đ?‘Ž and đ?‘?′ are projected onto đ?›˝ after the same line đ?‘?. Line đ?‘? is projected onto đ?›˝ after đ?‘?1 and đ?‘?1 âˆĽ đ?‘?. If đ?‘?1 âˆŚ đ?‘?, absurd because đ?‘?||đ?›ź(đ?‘?||đ?‘?′).

Solution to Problem 85.

đ?‘€ is the centroid in ∆đ??´đ??śđ??ˇ â&#x;š |đ?‘€đ??ˇ|

â&#x;š |đ?‘€đ?‘ƒ| = 2 (1) 80

255 Compiled and Solved Problems in Geometry and Trigonometry

đ?‘ is the centroid in ∆đ??´đ??ľđ??ˇ â&#x;š |đ?‘ đ??ˇ|

â&#x;š |đ?‘ đ?‘„| = 2 (2) đ??ż is the centroid in ∆đ??ľđ??śđ??ˇ â&#x;š â&#x;š

|đ??żđ??ˇ| |đ??żđ?‘†|

= 2 (3)

From 1 and 2, and from 2 and 3

because: So

Solution to Problem 86.

đ??ľđ??ˇ ⊼ (đ??´đ??´â€˛đ??ś) from the hypothesis đ??´đ??ľđ??śđ??ˇđ??´â€˛đ??ľâ€˛đ??śâ€˛đ??ˇâ€˛ cube (1). đ??´1 midpoint of segment |đ??ľđ??´â€˛| (đ??´đ??ľđ??´â€˛) isosceles and đ??´đ??´1 ⊼ đ??ľđ??´â€˛} |đ??´1 đ??´3 | mid-side in ∆đ??´â€˛đ??ľđ??ˇ â&#x;š đ??´1 đ??´3 âˆĽ đ??ľđ??ˇ (2) đ??´3 midpoint of |đ??´â€˛đ??ˇ| From (1) and (2) â&#x;š đ??´1 đ??´3 âˆĽ (đ??´đ??´â€˛đ??ś) â&#x;š đ??´â€˛đ??ś ⊼ đ??´1 đ??´3 (3) 81

Florentin Smarandache

From ∆đ??´đ??śđ??´â€˛:

From ∆đ??´đ??ľđ??´â€˛:

Similarly

In ∆đ??´đ??śđ??´â€˛:

and

đ??´1 đ??´2 đ??´â€˛ right with đ?‘š(đ??´â€˛đ??´2 đ??´1 ) = 90 because

From (4) and (3) â&#x;š đ??´â€˛đ??ś ⊼ (đ??´1 đ??´2 đ??´3 ). đ??´â€˛đ??ś ⊼ (đ??´1 đ??´2 đ??´3 ) As ′ } â&#x;š đ??´1 đ??´2 đ??´3 đ??´ coplanar â&#x;š đ??´1 đ??´2 đ??´3 đ??´ quadrilateral with đ??´ đ??ś ⊼ đ??´2 đ??´ (by construction) opposite angles đ??´1 and đ??´3 right â&#x;š đ??´1 đ??´2 đ??´3 đ??´ inscribable quadrilateral.

Solution to Problem 87. The conclusion is true only if ||đ??ľđ??ˇ|| = ||đ??´đ??ś|| that is đ?‘? = đ?‘?.

đ?‘€đ?‘ ⊼ đ??ˇđ??ś if

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255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 88.

bisector plane

(đ?‘? bisector half-plane) In triangle đ??ˇđ??ˇ1 đ??ˇâ€˛:

But 83

Florentin Smarandache

From 1, 2, 3 â&#x;š

đ?œŽ[đ??´đ??ľđ??ˇ] đ?œŽ[đ??´đ??ľđ??ś]

=

‖đ??ˇđ??¸â€– ‖đ??¸đ??śâ€–

q.e.d.

Solution to Problem 89.

Because the tetrahedron is regular đ??´đ??ľ = ‌ =

we increase the denominator

If one of the points đ?‘ƒ or đ?‘„ is on face đ??śđ??ľđ??ˇ the problem is explicit.

Solution to Problem 90. We consider tetrahedron đ?‘‚đ?‘Ľđ?‘Śđ?‘§, and prove that the sum of the measures of the dihedral angles of this trihedron is bigger than 360°. Indeed: let 100′ be the internal bisector of trihedron đ?‘‚đ?‘Ľđ?‘Śđ?‘§ (1000′ the intersection of the bisector planes of the 3 dihedral angles) of the trihedron in đ??´, đ??ľ, đ??ś.

84

255 Compiled and Solved Problems in Geometry and Trigonometry

The size of each dihedron with edges đ?‘œđ?‘Ľ, đ?‘œđ?‘Ś, đ?‘œđ?‘§ is bigger than the size of the corresponding angles of đ??´đ??ľđ??ś, the sum of the measures of the dihedral angles of trihedron đ?‘‚đ?‘Ľđ?‘Śđ?‘§ is bigger than 180°. Let (đ?‘Ž, đ?‘?) be a plane ⊼ to đ?‘œđ?‘§ at đ??ś; đ?‘Ž ⊼ đ?‘œđ?‘§, đ?‘? ⊼ đ?‘œđ?‘§, but |đ??śđ??´ and |đ??śđ??ľ are on the same Ě‚ ). half-space in relation to (đ?‘Žđ?‘?) ⇒ đ?‘š(đ??śĚ‚ ) < đ?‘š(đ?‘Žđ?‘? In tetrahedron đ??´đ??ľđ??śđ??ˇ, let đ?‘Ž1 , đ?‘Ž2 , đ?‘Ž3 , đ?‘Ž4 , đ?‘Ž5 and đ?‘Ž6 be the 6 dihedral angles formed by the faces of the tetrahedron.

according to the inequality previously established.

Solution to Problem 91.

We mark with a the intersection of planes (đ??´, đ?‘‘1 ) and (đ??ľ, đ?‘‘2 ). So Let đ?‘? be a variable line that passes through đ??ś and contained in đ?›ź, which cuts đ?‘‘1 and đ?‘‘2 at đ?‘€ respectively đ?‘ . We have: đ?‘€đ??´ ⊂ (đ??´, đ?‘‘1 ), đ?‘€đ??´ ∊ đ?‘ đ??ľ = đ?‘ƒ(đ?‘€đ??´ and đ?‘ đ??ľ intersect because they are contained in the plane determined by (đ??´đ?‘€, đ?‘?)). Thus đ?‘ƒ ∈ (đ??´, đ?‘‘1 ) and đ?‘ƒ ∈ (đ??ľ, đ?‘‘2 ), â&#x;š đ?‘ƒ ∈ đ?‘Ž, so đ?‘ƒ describes line a the intersection of planes (đ??´, đ?‘‘1 ) and (đ??ľ, đ?‘‘2 ).

Vice-versa: let đ?‘„ ∈ đ?‘Ž. In the plane (đ??´, đ?‘‘1 ): đ?‘„đ??´ ∊ đ?‘‘1 = đ?‘€â€˛ In the plane (đ??ľ, đ?‘‘2 ): đ?‘„đ??ľ ∊ đ?‘‘2 = đ?‘ ′ 85

Florentin Smarandache

Lines đ?‘ ′đ?‘€â€˛ and đ??´đ??ľ are coplanar (both are on plane (đ?‘„, đ??´, đ??ľ)). But because đ?‘ ′đ?‘€â€˛ ⊂ đ?›ź and đ??´đ??ľ has only point đ??ś in common with đ?›ź â&#x;š đ?‘€â€˛đ?‘ ′ ∊ đ??´đ??ľ = đ??ś. So đ?‘€â€˛đ?‘ ′ passes through đ??ś. If planes (đ??´, đ?‘‘1 ) and (đ??ľ, đ?‘‘2 ) are parallel, the locus is the empty set.

Solution to Problem 92.

Remember the theorem: If a plane đ?›ž intersects two planes đ?›ź and đ?›˝ such that đ?œŽ||đ?›ź â&#x;š (đ?›ž ∊ đ?›ź)||(đ?›ž ∊ đ?›˝). If plane (đ??żđ?‘€đ?‘ đ?‘ƒ)||đ??ľđ??ˇ we have:

If (đ??żđ?‘€đ?‘ đ?‘ƒ)||đ??´đ??ś we have:

â&#x;š relation đ?‘Ž.

Solution: Let đ??´â€˛, đ??ľâ€˛, đ??śâ€˛, đ??ˇâ€˛ the projections of points đ??´, đ??ľ, đ??ś, đ??ˇ onto plane (đ?‘€đ?‘ đ?‘ƒđ??ż). For ex. points đ??ľâ€˛, đ??ż, đ??´â€˛ are collinear on plane (đ??żđ?‘ƒđ?‘€đ?‘ ) because they are on the projection of line đ??´đ??ľ onto this plane.

Similarly we obtain:

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255 Compiled and Solved Problems in Geometry and Trigonometry

By multiplying the 4 relations,

â&#x;š relation (đ?‘Ž) from đ?‘‘.

Solution to Problem 93.

đ?‘€ – Midpoint of |đ??ľđ??ś|.

Solution to Problem 94.

(1) đ??´đ??ľ ⊼ đ??śđ??ˇ (hypothesis)

From 1 and 2

height in ∆đ??´đ??ľđ??ś a

From 3 and 4 đ??´đ??ś ⊼ (đ??ľđ??ˇđ??ť) â&#x;š đ??´đ??ś ⊼ đ??ľđ??ť â&#x;š đ??ľđ??ť height in ∆đ??´đ??ľđ??ś

b

From a and b â&#x;š đ??ť orthocenter ∆đ??´đ??ľđ??ś. Let đ??ś1 be the diametrical opposite point to 0 đ??ś in circle đ??ś(đ??´đ??ľđ??ś)đ??śđ??ś1 diameter đ?‘š(đ??śĚ‚ 1 đ??ľđ??ś ) = 90 but đ??´đ??ť ⊼ đ??ľđ??ś â&#x;š đ??´đ??ť âˆĽ đ??ľđ??ś1 . Similarly

đ??ľđ??ť||đ??ś1 đ??´, so đ??´đ??ťđ??ľđ??ś1 parallelogram, we have: 87

Florentin Smarandache

similarly

diametrical opposite to B

but

by substituting above, we have:

Let đ?‘ , đ?‘€, đ?‘„, đ?‘ƒ, đ?‘†, đ?‘… midpoints of the edges of the quadrilateral đ?‘ đ?‘€đ?‘ƒđ?‘„ because: đ?‘ đ?‘€||đ??śđ??ˇ||đ?‘ƒđ?‘„ (median lines), đ?‘„đ?‘€||đ??´đ??ľ||đ?‘ƒđ?‘ (median lines), but đ??śđ??ˇ ⊼ đ??´đ??ľ ⇒ đ?‘€đ?‘ đ?‘„đ?‘ƒ rectangle |đ?‘ đ?‘„| ∊ |đ?‘ƒđ?‘€| = {0}. Similarly đ?‘€đ?‘†đ?‘ƒđ?‘… rectangle with |đ?‘€đ?‘ƒ| common diagonal with a, the first rectangle, so the 6 points are equally distant from “đ?‘‚â€? the midpoint of diagonals in the two rectangles â&#x;š the 6 points are on a sphere.

Solution to Problem 95.

đ?‘€ arbitrary point on |đ??´đ??ľâ€˛| đ??´Ęş midpoint of segment [đ??ľâ€˛đ??śâ€˛] When đ?‘€ = đ??ľâ€˛, point đ?‘ƒ is in the position đ??ľâ€˛.

88

255 Compiled and Solved Problems in Geometry and Trigonometry

When đ?‘€ = đ??´, point đ?‘ƒ is in the position {đ?‘ƒ1 } = [đ??´â€˛đ??´1 ] ∊ [đ??´đ??´â€˛â€˛] (đ??´â€˛đ??´â€˛â€˛đ??´1 đ??´ rectangle, so đ?‘ƒ1 is the intersection of the diagonals of the rectangle) [The locus is [đ??ľâ€˛đ?‘ƒ1 ]]. Let đ?‘€ be arbitrary point đ?‘€ ∈ |đ??´đ??ľâ€˛|. because: By the way it was constructed â&#x;š (∀) plane that contains đ??´đ??´1 is perpendicular to (đ??ľâ€˛đ??śđ??śâ€˛â€˛), particularly to (đ??ľâ€˛đ??´đ??´1 ) ⊼ (đ??ľâ€˛đ??śâ€˛đ??ś). because đ??ľâ€˛, đ?‘ƒ ∈ (đ??ľâ€˛đ??´đ??´â€˛â€˛) from this reason đ??ľâ€˛, đ?‘ƒ1 ∈ (đ??´â€˛đ??ľâ€˛đ??´1 ). From 1 and 2 Let

So (∀) đ?‘€ ∈ |đ??ľâ€˛đ??´| and we have

Vice-versa. Let đ?‘ƒ arbitrary point, đ?‘ƒ ∈ |đ??ľâ€˛đ?‘ƒ1 | and In plane

In plane Indeed: đ??´â€˛đ??´â€˛â€˛ || (đ??ľâ€˛đ??´đ??´1 ) thus any plane which passes through đ??´â€˛đ??´â€˛â€˛ will intersect (đ??ľâ€˛đ??´đ??´1 ) after a parallel line to đ??´â€˛đ??´â€˛â€˛. Deci đ?‘€đ?‘ ||đ??´â€˛đ??´â€˛â€˛ or đ?‘€đ?‘ ||đ??´đ??´1 as đ?‘€ ∈ (đ??ľâ€˛đ??´đ??´1 ) â&#x;š đ?‘€đ?‘ ⊼ (đ??ľâ€˛đ??śđ??śâ€˛â€˛). We’ve proved

89

Florentin Smarandache

and we have describes |đ??ľâ€˛đ?‘ƒ1 | and vice-versa, there is đ?‘€|đ??ľâ€˛đ??´| and đ?‘ |đ??ľâ€˛đ??´1 | such that and đ?‘ƒ is the intersection of the diagonals of the quadrilateral đ??´â€˛đ?‘ đ?‘€đ??´â€˛â€˛.

Solution to Problem 96.

known result

From 1 and 3

Solution to Problem 97. From the hypothesis: đ?‘‚đ??´ ⊼ đ?‘‚đ??ľ ⊼ đ?‘‚đ??ś ⊼ đ?‘‚đ??´ We assume the problem is solved. Let đ?‘€ be the orthocenter of triangle đ??´đ??ľđ??ś. 90

255 Compiled and Solved Problems in Geometry and Trigonometry

𝐴𝐵 ⊥ (𝑂𝐶𝐶′) } ⟹ 𝐴𝐵 ⊥ 𝑂𝑀 ⟹ 𝑀𝑂 ⊥ 𝐴𝐵 (1) 𝑀𝑂 ⊥ (𝐶𝑂𝐶′) 𝐵𝐶 ⊥ (𝐴𝑂𝐴′) 𝐴𝑂 ⊥ (𝐶𝑂𝐵) ⟹ 𝐴𝑂 ⊥ 𝐵𝐶, but 𝐴𝐴′ ⊥ 𝐵𝐶 ⟹ } ⟹ 𝐵𝐶 ⊥ 𝑀𝑂 ⟹ 𝑀𝑂 ⊥ 𝐵𝐶 (2) 𝑀𝑂 ⊂ (𝐴𝑂𝐴′) But 𝐶𝐶′ ⊥ 𝐴𝐵 ⟹

From (1) and (2) ⟹ 𝑀𝑂 ⊥ (𝐴𝐵𝐶) So the plane (𝐴𝐵𝐶) that needs to be drawn must be perpendicular to 𝑂𝑀 at 𝑀.

Solution to Problem 98.

𝐴′𝑁 ⊥ 𝐴𝐷, 𝐵′𝑀 ⊥ 𝐵𝐶 ‖𝐵𝑀‖ =

𝑎−𝑎′ , 2

𝑣[𝐵𝑀𝑃𝑆𝐵′] =

‖𝑃𝑀‖ =

𝑏−𝑏′ 2

𝑎 − 𝑎′ 𝑏 − 𝑏′ ℎ ∙ ∙ 2 2 3

𝑣[𝑆𝑃𝑊𝑅𝐴′𝐵′] =

𝜎[𝑆𝑃𝐵′] ∙ ‖𝐵′𝐴′‖ 𝑎 − 𝑎′ ℎ ′ = ∙ ∙𝑏 3 2 2

𝑣[𝐵′𝐴′𝑁𝑀𝐶′𝐷′𝐷1 𝐶1 ] =

(𝑏 + 𝑏′)ℎ ∙ 𝑎′ 2

𝑣[𝐴𝐵𝐴′𝐵′𝐶𝐷𝐶′𝐷′] = 2 [2 ∙ ℎ 6

𝑎−𝑎′ 2

∙

𝑏−𝑏′ 2

ℎ

𝑎−𝑎′

3

2

∙ +

ℎ

(𝑏+𝑏′)ℎ

2

2

∙ ∙ 𝑏′] + (

ℎ

) ∙ 𝑎′ =

(2𝑎𝑏 − 2𝑎𝑏 ′ − 2𝑎′ 𝑏 ′ + 3𝑎𝑏 ′ − 3𝑎′ 𝑏 ′ + 3𝑎′ 𝑏 + 3𝑎′ 𝑏 ′ ) = [𝑎𝑏 + 𝑎′ 𝑏 ′ + 6

(𝑎 + 𝑎′)(𝑏 + 𝑏′)].

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Florentin Smarandache

Solution to Problem 99.

𝐵∙ℎ 2 ℎ 𝐵ℎ 𝑏ℎ 𝑣[𝑂𝐴′𝐵′𝐴] = 𝑣[𝐴𝐵𝑂𝑂′𝐴′𝐵′] − 𝑣[𝐴𝐵𝐵′ 𝑂] − 𝑣[𝐴′ 𝐵′ 𝑂′ 𝑂] = (𝐵 + 𝑏 + √𝐵𝑏) − − = 𝑣[𝑂𝐴𝐵𝐵′] =

3

3

ℎ √𝑏𝐵. 3

So: ℎ 𝑣[𝑂𝐴′𝐵′𝐴] 3 √𝐵𝑏 ∙ √𝐵𝑏 √𝑏 √𝑏 = = ⟹ 𝑣[𝑂𝐴′𝐵′𝐴] = ∙ 𝑣[𝑂𝐴𝐵𝐵′] 𝐵ℎ 𝑣[𝐷𝐴𝐵𝐵′] √𝐵 √𝐵 ∙ 𝐵 3 For the relation above, determine the formula of the volume of the pyramid frustum.

Solution to Problem 100.

𝑑(𝐺𝐺′) = ℎ = 2𝑅 Let 𝑙 = ‖𝐴𝐶‖ ⟹ ‖𝐴𝐷‖ =

𝑙 √3 2

⟹ ‖𝐺𝐷‖ =

𝑙 √3 6

Figure 𝐺𝐷𝑀𝑂 rectangle ⟹ ‖𝐺𝐷‖ = ‖𝑂𝑀‖ ⟹

92

𝑙 √3 6

= 𝑅 = 2√3𝑅

3

255 Compiled and Solved Problems in Geometry and Trigonometry

So, the lateral area is đ?‘†đ?‘™ = 3 ∙ 2√3đ?‘… ∙ đ?‘… = 12√3đ?‘…2. đ?‘Ł[đ??´đ??ľđ??śđ??´â€˛đ??ľâ€˛đ??śâ€˛] = đ?œŽ[đ??´đ??ľđ??ś] ∙ 2đ?‘… = 2√3đ?‘… ∙

2√3đ?‘…√3 4

∙ 2đ?‘… = 6√3đ?‘… 2.

The total area: đ?‘†đ?‘Ą = đ?‘†đ?‘™ + 2đ?œŽ[đ??´đ??ľđ??ś] = 12√3đ?‘…2 + 2 ∙ 3đ?‘…√3đ?‘… 2 = 18√3đ?‘…2

Solution to Problem 101.

Let đ?‘‰1 and đ?‘†1 be the volume, respectively the area obtained revolving around đ?‘Ž. đ?‘‰2 and đ?‘†2 be the volume, respectively the area obtained after revolving around đ?‘?. đ?‘‰3 and đ?‘†3 be the volume, respectively the area obtained after revolving around c. So: đ?‘‰1 =

đ?œ‹ ∙ đ?‘– 2 (‖đ??śđ??ˇâ€– + ‖đ??ˇđ??ľâ€–) đ?œ‹ ∙ đ?‘– 2 ∙ đ?‘Ž = 3 3

đ?‘†1 = đ?œ‹ ∙ đ?‘– ∙ đ?‘? + đ?œ‹ ∙ đ?‘– ∙ đ?‘? = đ?œ‹ ∙ đ?‘– ∙ (đ?‘? + đ?‘?) đ?‘‰2 =

đ?œ‹đ?‘? 2 đ?‘? đ?œ‹đ?‘? 2 đ?‘?2 đ?œ‹đ?‘? 2 đ?‘? 2 đ?‘Ž = = 3 3đ?‘? 3đ?‘Ž2 đ?‘†2 = đ?œ‹ ∙ đ?‘? ∙ đ?‘Ž

�3 =

đ?œ‹đ?‘? 2 đ?‘? đ?œ‹đ?‘? 2 đ?‘? 2 đ?œ‹đ?‘? 2 đ?‘? 2 = = 3 3đ?‘? 3đ?‘Ž đ?‘†3 = đ?œ‹ ∙ đ?‘? ∙ đ?‘Ž

Therefore: 1 1 1 9đ?‘Ž2 9đ?‘? 2 9đ?‘? 2 = + â&#x;ş = + (đ?œ‹đ?‘? 2 đ?‘? 2 )2 (đ?œ‹đ?‘? 2 đ?‘? 2 )2 (đ?œ‹đ?‘? 2 đ?‘? 2 )2 đ?‘‰12 đ?‘‰22 đ?‘‰32 đ?‘†2 đ?‘†3 đ?‘†2 + đ?‘†3 đ?‘? đ?‘? đ?œ‹đ?‘Ž(đ?‘? + đ?‘?) đ?‘? 2 + đ?‘?2 đ?‘Ž + = â&#x;ş + = â&#x;ş = đ?‘†3 đ?‘†2 đ?‘†1 đ?‘? đ?‘? đ?œ‹đ?‘–(đ?‘? + đ?‘?) đ?‘?∙đ?‘? đ?‘– But đ?‘– ∙ đ?‘Ž = đ?‘? ∙ đ?‘? â&#x;š

đ?‘? 2 +đ?‘?2 đ?‘?đ?‘?

đ?‘Ž2

= đ?‘?đ?‘? , ‖đ??´đ??ˇâ€– = đ?‘–.

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Florentin Smarandache

Solution to Problem 102.

𝑟 = ‖𝑂𝐴‖ = 25 𝑐𝑚 𝑅 = ‖𝑂′𝐵‖ = 50 𝑐𝑚 2𝜋𝑟 = 1,57 ⟹ 𝑟 = 0,25 𝑚 2𝜋𝑅 = 3,14 ⟹ 𝑅 = 0,50 𝑚 ‖𝐶𝑁‖ = 18 𝑐𝑚 = 0,18 𝑚 ‖𝐴′𝐵‖ = 25 𝑐𝑚 ‖𝐴𝐵‖ = √100 + 0,0625 = 10,003125 ‖𝐴′𝑀‖ =

‖𝐴𝐴′‖ ∙ ‖𝐴′𝐵‖ 10 ∙ 0,25 = ≈ 0,25 ‖𝐴𝐵‖ 10,003125

‖𝐶𝑁‖ ‖𝐶𝐵‖ 0,18 ‖𝐶𝐵‖ = ⟹ = ⟹ ‖𝐶𝐵‖ = 0,18 ‖𝐴′𝑀‖ ‖𝐵𝐴′‖ 0,25 0,25 ‖𝑂′𝐶‖ = 𝑅 ′ = 0,50 − 0,18 = 0,32 ‖𝑂𝑃‖ = 𝑟 ′ = 0,25 − 0,18 = 0,07 𝑉= 𝑉=

𝜋1 2 (𝑅 + 𝑟 2 + 𝑅𝑟) 3

𝜋10 (0,502 + 0,252 + 0,50 ∙ 0,25 − 0,322 − 0,072 − 0,32 ∙ 0,07) 3 𝜋10 (0,4375 − 0,1297) = 1,026𝜋𝑚3 = 3

Solution to Problem 103. ‖𝑉𝑃‖ = In ∆𝑉𝐴𝑃: ‖𝑉𝐴‖ =

𝐴 𝛼 2

2 sin

𝑎

𝛼 𝛼 cos 2 2 sin 2

.

94

255 Compiled and Solved Problems in Geometry and Trigonometry

In ∆đ?‘‰đ??´đ?‘‚′: ‖đ?‘‰đ?‘‚′‖2 =

đ?‘Ž2 4 sin2

� 2

−

đ?‘Ž2 2

. ‖��′‖ =

In ∆đ?‘‰đ?‘‚đ?‘‚′: ‖đ?‘‚đ?‘‚′‖ =

đ?‘Žâˆšcos đ?›ź đ?›ź 2

2 sin

đ?‘Žâˆšcos đ?›ź đ?›ź 2 sin 2

− đ?‘…. 2

đ?›ź 2 2 4đ?‘…√cos đ?›ź sin2 2 đ?‘Ž đ?‘Ž cos đ?›ź đ?‘Ž đ?‘Ž cos đ?›ź 2đ?‘Žđ?‘… cos đ?›ź √ √ đ?‘…2 = +( đ?›ź − đ?‘…) â&#x;š 2 + đ?›źâˆ’ đ?›ź â&#x;šđ?‘Ž= đ?›ź đ?›ź 2 2 sin 2 4 sin2 2 2 sin 2 sin 2 (2 cos 2 2 + cos đ?›ź) đ?›ź â&#x;š đ?‘Ž = 4đ?‘…√cos đ?›ź ∙ sin 2 2

đ?›ź đ?›ź đ?›ź đ?‘Ž2 cos 2 đ?‘Ž2 cos 2 đ?›ź cos 2 2 2 2 2 đ??´đ?‘™ = 4 đ?›ź= đ?›ź = 16đ?‘… cos đ?›ź sin 2 ∙ đ?›ź = 8đ?‘… cos đ?›ź sin đ?›ź = 4đ?‘… sin 2đ?›ź 2 ∙ 2 sin 2 sin 2 sin 2 đ?›ź đ??´đ?‘Ą = đ??´đ?‘™ + đ?‘Ž2 = 4đ?‘… 2 sin 2đ?›ź + 16đ?‘… 2 cos đ?›ź sin2 2

‖đ?‘‰đ?‘‚′‖ = đ?‘… â&#x;š đ?‘… =

đ?‘Žâˆšcos đ?›ź đ?›ź √cos đ?›ź 0 đ?‘Ž â&#x;š đ?‘… = 4đ?‘…√cos đ?›ź sin 2 ∙ đ?›ź â&#x;š 2 cos đ?›ź = 1 â&#x;š đ?›ź = 60 . 2 sin 2 2 sin 2

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Various Problems 104. Determine the set of points in the plane, with affine coordinates đ?‘§ that satisfy: a. |đ?‘§| = 1; b. đ?œ‹ < arg đ?‘§ ≤ c. arg đ?‘§ >

4đ?œ‹ 3

3đ?œ‹ 2

; � ≠0;

, � ≠0;

d. |� + �| ≤ 2 . Solution to Problem 104

105. Prove that the đ?‘› roots of the unit are equal to the power of the particular root đ?œ€1 . Solution to Problem 105

106. Knowing that complex number đ?‘§ verifies the equation đ?‘§ đ?‘› = đ?‘›, show that numbers 2, −đ?‘–đ?‘§ and đ?‘–đ?‘§ verify this equation.

Application: Find (1 − 2đ?‘–)4 and deduct the roots of order 4 of the number −7 + 24đ?‘–. Solution to Problem 106

107. Show that if natural numbers đ?‘š and đ?‘› are coprime, then the equations đ?‘§ đ?‘š − 1 = 0 and đ?‘§ đ?‘› − 1 = 0 have a single common root. Solution to Problem 107

108. Solve the following binomial equation: (2 − 3đ?‘–)đ?‘§ 6 + 1 + 5đ?‘– = 0. Solution to Problem 108

109. Solve the equations:

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255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 109

110. Solve the equation đ?‘§Ě… = đ?‘§ đ?‘›âˆ’1 , đ?‘› ∈ đ?‘ , where đ?‘§Ě… the conjugate of đ?‘§. Solution to Problem 110

111. The midpoints of the sides of a quadrilateral are the vertices of a parallelogram. Solution to Problem 111

112. Let đ?‘€1 đ?‘€2 đ?‘€3 đ?‘€4 and đ?‘ 1 đ?‘ 2 đ?‘ 3 đ?‘ 4 two parallelograms and đ?‘ƒđ?‘– the midpoints of segments [đ?‘€đ?‘– đ?‘ đ?‘– ], đ?‘– ∈ {1, 2, 3, 4}. Show that đ?‘ƒ1 đ?‘ƒ2 đ?‘ƒ3 đ?‘ƒ4 is a parallelogram or a degenerate parallelogram. Solution to Problem 112

113. Let the function đ?‘“: đ??ś → đ??ś, đ?‘“(đ?‘§) = đ?‘Žđ?‘§ + đ?‘?; (đ?‘Ž, đ?‘?, đ?‘? ∈ đ??ś, đ?‘Ž ≠0). If đ?‘€1 and đ?‘€2 are of affixes đ?‘§1 and đ?‘§2 , and đ?‘€1′ and đ?‘€2′ are of affixes đ?‘“(đ?‘§1 ), đ?‘“(đ?‘§2 ), show that ‖đ?‘€1′ đ?‘€2′ ‖ = |đ?‘Ž| ∙ ‖đ?‘€1 đ?‘€2 ‖. We have ‖đ?‘€1′ đ?‘€2′ ‖ = ‖đ?‘€1 đ?‘€2 ‖ ⇔ |đ?‘Ž| = 1. Solution to Problem 113

114. Prove that the function z → zĚ…, z ∈ C defines an isometry. Solution to Problem 114

115. Let M1 M2 be of affixes đ?‘§1 , đ?‘§2 ≠0 and z2 = Îąz1 . Show that rays |OM1, |OM2 coincide (respectively are opposed) â&#x;ş Îą > 0 (respectively Îą < 0). Solution to Problem 115

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116. Consider the points M1 M2 M3 of affixes đ?‘§1 đ?‘§2 đ?‘§3 and đ?‘€1 ≠đ?‘€2 . Show that: z −z

a. M3 ∈ |M1 M2 â&#x;ş z3 −z1 > 0; 2

1

z3 −z1

b. M3 ∈ M1 M2 â&#x;ş z

2 −z1

∈R. Solution to Problem 116

117. Prove Pompeiu’s theorem. If the point đ?‘€ from the plane of the equilateral triangle đ?‘€1 đ?‘€2 đ?‘€3 ∉ the circumscribed circle ∆ đ?‘€1 đ?‘€2 đ?‘€3 ďƒž there exists a triangle having sides of length ‖đ?‘€đ?‘€1 ‖, ‖đ?‘€đ?‘€2 ‖, ‖đ?‘€đ?‘€3 ‖. Solution to Problem 117

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255 Compiled and Solved Problems in Geometry and Trigonometry

Solutions Solution to Problem 104. a.

|đ?‘§| = 1 |đ?‘§| = √đ?‘Ľ 2 + đ?‘Ś 2

b. đ?œ‹ < argđ?‘§ ≤

3đ?œ‹ 2

} â&#x;š đ?‘Ľ 2 + đ?‘Ś 2 = 1, so the desired set is the circle đ??ś(0,1).

.

The desired set is given by all the points of quadrant III, to which ray |đ?‘‚đ?‘Ś is added, so all the points with đ?‘Ľ < 0, đ?‘Ś < 0. c.

arg� >

4đ?œ‹ 3

,� ≠0

argđ?‘§ ∈ [0, 2đ?œ‹]

}â&#x;š

4đ?œ‹ 2

< argđ?‘§ < 2đ?œ‹

The desired set is that of the internal points of the angle with its sides positive semi-axis and ray |đ?‘‚đ??ľ. d. |đ?‘§ + đ?‘–| ≤ 2; đ?‘§ = đ?‘Ľ + đ?‘Śđ?‘–, its geometric image đ?‘€.

where đ?‘‚′ (0, −1). ′ Thus, the desired set is the disk centered at đ?‘‚(0,−1) and radius 2.

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Solution to Problem 105.

Solution to Problem 106. Let the equation đ?‘§ 4 = đ?‘›. If đ?‘§ 4 = 4 (đ?‘§ is the solution) then: (−đ?‘§)4 = (−1)4 đ?‘§ 4 = 1 ∙ đ?‘› = đ?‘›, so – đ?‘§ is also a solution.

â&#x;š đ?‘–đ?‘§ is the solution;

â&#x;š −đ?‘–đ?‘§ is the solution;

â&#x;š is the solution of the equation đ?‘§ 4 = −7 + 24đ?‘–. The solutions of this equation are:

but based on the first part, if đ?‘§ − 1 − 2đ?‘– is a root, then

are solutions of the given equation.

Solution to Problem 107.

If there exist đ?‘˜ and đ?‘˜â€˛ with đ?‘§đ?‘˜ = đ?‘§đ?‘˜â€˛ , then

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255 Compiled and Solved Problems in Geometry and Trigonometry

because (đ?‘š, đ?‘›) = 1. Because đ?‘˜ ′ < đ?‘›, đ?‘˜ < đ?‘š, we have đ?‘˜â€˛ = 0, đ?‘˜ = 0. Thus the common root is đ?‘§0 .

Solution to Problem 108.

Solution to Problem 109.

Solution to Problem 110.

As:

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From:

positive

The given equation becomes

Solution to Problem 111.

We find the sum of the abscissa of the opposite points:

â&#x;š đ?‘€đ?‘ đ?‘ƒđ?‘„ a parallelogram.

Solution to Problem 112. In the quadrilateral đ?‘€1 đ?‘€3 đ?‘ 3 đ?‘ 1 by connecting the midpoints we obtain the parallelogram đ?‘‚′ đ?‘ƒ1 đ?‘‚′′ đ?‘ƒ3 , with its diagonals intersecting at đ?‘‚, the midpoint of |đ?‘‚′đ?‘‚′′| and thus |đ?‘ƒ1 đ?‘‚| ≥ |đ?‘‚đ?‘ƒ3 |. (1) 102

255 Compiled and Solved Problems in Geometry and Trigonometry

In the quadrilateral đ?‘€4 đ?‘€2 đ?‘ 2 đ?‘ 4 by connecting the midpoints of the sides we obtain the parallelogram đ?‘‚′ đ?‘ƒ2 đ?‘‚′′ đ?‘ƒ4 with its diagonals intersecting in đ?‘‚, the midpoint of |đ?‘‚′đ?‘‚′′| and thus |đ?‘ƒ2 đ?‘‚| ≥ |đ?‘‚đ?‘ƒ4 |. (2)

From (1) and (2) đ?‘ƒ1 đ?‘ƒ2 đ?‘ƒ3 đ?‘ƒ4 a parallelogram.

Solution to Problem 113.

If:

If:

Solution to Problem 114. Let �1 and �2 be of affixes �1 and �2 . Their images through the given function �1′ and �2′ with affixes �̅1 and �̅2 , so

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From (1) and (2) â&#x;š ‖đ?‘€1 đ?‘€2 ‖ = ‖đ?‘€1′ đ?‘€2′ ‖ or ‖đ?‘€1′ đ?‘€2′ ‖ = |đ?‘§Ě…2 đ?‘§Ě…1 | = |√đ?‘§2 − đ?‘§1 | = |đ?‘§2 − đ?‘§1 | = ‖đ?‘€1 đ?‘€2 ‖. So đ?‘“: đ??ś → đ??ś, đ?‘“(đ?‘§) = đ?‘§Ě… defines an isometry because it preserves the distance between the points.

Solution to Problem 115. We know that the argument (đ?‘Žđ?‘§1 ) = argđ?‘§1 + argđ?‘§đ?›ź − 2đ?‘˜đ?œ‹, where đ?‘˜ = 0 or đ?‘˜ = 1. Because argđ?‘§2 = arg(đ?‘Žđ?‘§1 ), argđ?‘§2 = argđ?‘§1 + argđ?‘§đ?›ź − 2đ?‘˜đ?œ‹. a. We assume that

Vice versa,

b. Let |đ?‘‚đ?‘€1 and |đ?‘‚đ?‘€2 be opposed â&#x;š argđ?‘§2 = argđ?‘§1 + đ?œ‹

∈ to the negative ray |đ?‘‚đ?‘Ľ ′ â&#x;š đ?›ź < 0. Vice versa,

đ?‘˜ = 0 or đ?‘˜ = 1 â&#x;š

argđ?‘§2 = argđ?‘§1 + đ?œ‹ or } â&#x;š |đ?‘‚đ?‘€1 and |đ?‘‚đ?‘€2 are opposed. argđ?‘§2 = argđ?‘§1 − đ?œ‹

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255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 116. If n and đ?‘›â€˛ are the geometric images of complex numbers đ?‘§ and đ?‘§â€˛, then the image of the difference đ?‘§â€“ đ?‘§â€˛ is constructed on |OM1 | and |đ?‘€â€˛đ?‘€| as sides. We assume that đ?‘€3 ∈ |đ?‘€1 đ?‘€2 We construct the geometric image of đ?‘§2 – đ?‘§1 . It is the fourth vertex of the parallelogram đ?‘‚đ?‘€1 đ?‘€2 đ?‘„1 . The geometric image of đ?‘§3 – đ?‘§1 is đ?‘„2 , the fourth vertex of the parallelogram đ?‘‚đ?‘€1 đ?‘€3 đ?‘„2 .

đ?‘‚đ?‘„1 âˆĽ đ?‘€1 đ?‘€2 đ?‘‚đ?‘„2 âˆĽ đ?‘€1 đ?‘€3 đ?‘€1 đ?‘€2 đ?‘€3 collinear

} â&#x;š đ?‘„1 , đ?‘„2 , đ?‘„3 collinear â&#x;š

Vice versa, we assume that

If đ?‘€3 and đ?‘€2 ∈ the opposite ray to đ?‘‚, then đ?‘§3 – đ?‘§1 = đ?›ź(đ?‘§2 – đ?‘§1 ) with đ?›ź < 0. We repeat the reasoning from the previous point for the same case. 105

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Thus, when đ?‘€3 ∈ đ?‘€1 đ?‘€2 đ?‘€3 + đ?‘€2 we obtain for the respective ratio positive, negative or having đ?‘€3 = đ?‘€1, so

z3 −z1 z2 −z1

∈ R.

Solution to Problem 117. The images of the roots of order 3 of the unit are the peaks of the equilateral triangle.

But É›1 = É›22 , so if we write É›2 = đ?œ€, then É›1 = É›2 . Thus đ?‘€1 (1), đ?‘€2 (đ?œ€), đ?‘€3 (đ?œ€ 2 ). We use the equality:

adequate (∀)đ?‘§ ∈ â„‚.

But

Therefore,

By substitution:

but

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255 Compiled and Solved Problems in Geometry and Trigonometry

thus

Therefore ‖đ?‘€đ?‘€1 ‖, ‖đ?‘€đ?‘€2 ‖, ‖đ?‘€đ?‘€3 ‖ sides of a ∆. Then we use ‖đ?‘Ľ| − |đ?‘Śâ€– ≤ |đ?‘Ľ − đ?‘Ś| and obtain the other inequality.

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Problems in Spatial Geometry

118. Show that if a line đ?‘‘ is not contained in plane đ?›ź, then đ?‘‘ ∊ đ?›ź is ∅ or it is formed of a single point. Solution to Problem 118

119. Show that (∀) đ?›ź, (∃) at least one point which is not situated in đ?›ź. Solution to Problem 119

120. The same; there are two lines with no point in common. Solution to Problem 120

121. Show that if there is a line đ?‘‘ (∃) at least two planes that contain line đ?‘‘. Solution to Problem 121

122. Consider lines �, �′, �′′, such that, taken two by two, to intersect. Show that, in this case, the 3 lines have a common point and are located on the same plane. Solution to Problem 122

123. Let đ??´, đ??ľ, đ??ś be three non-collinear points and đ??ˇ a point located on the plane (đ??´đ??ľđ??ś). Show that: a. The points đ??ˇ, đ??´, đ??ľ are not collinear, and neither are đ??ˇ, đ??ľ, đ??ś; đ??ˇ, đ??ś, đ??´. b. The intersection of planes (đ??ˇđ??´đ??ľ), (đ??ˇđ??ľđ??ś), (đ??ˇđ??śđ??´) is formed of a single point. Solution to Problem 123

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255 Compiled and Solved Problems in Geometry and Trigonometry

124. Using the notes from the previous exercise, take the points đ??¸, đ??š, đ??ş distinct from đ??´, đ??ľ, đ??ś, đ??ˇ, such that đ??¸ ∈ đ??´đ??ˇ, đ??š ∈ đ??ľđ??ˇ, đ??ş ∈ đ??śđ??ˇ. Let đ??ľđ??ś ∊ đ??šđ??ş = {đ?‘ƒ}, đ??şđ??¸ ∊ đ??śđ??´ = {đ?‘„}, đ??¸đ??š ∊ đ??´đ??ľ = {đ?‘…}. Show that đ?‘ƒ, đ?‘„, đ?‘… are collinear (T. Desarques). Solution to Problem 124

125. Consider the lines đ?‘‘ and đ?‘‘′ which are not located on the same plane and the distinct points đ??´, đ??ľ, đ??ś ∈ đ?‘‘ and đ??ˇ, đ??¸ ∈ đ?‘‘′. How many planes can we draw such that each of them contains 3 non-collinear points of the given points? Generalization. Solution to Problem 125

126. Show that there exist infinite planes that contain a given line đ?‘‘. Solution to Problem 126

127. Consider points đ??´, đ??ľ, đ??ś, đ??ˇ which are not located on the same plane. a. How many of the lines đ??´đ??ľ, đ??´đ??ś, đ??´đ??ˇ, đ??ľđ??ś, đ??ľđ??ˇ, đ??śđ??ˇ can be intersected by a line that doesn’t pass through đ??´, đ??ľ, đ??ś, đ??ˇ? b. Or by a plane that doesn’t pass through đ??´, đ??ľ, đ??ś, đ??ˇ? Solution to Problem 127

128. The points đ?›ź and đ?›˝ are given, đ??´, đ??ľ ∈ đ?›ź. Construct a point đ?‘€ ∈ đ?›ź at an equal distance from đ??´ and đ??ľ, that ∈ also to plan đ?›˝. Solution to Problem 128

129. Determine the intersection of three distinct planes �, �, �. Solution to Problem 129

130. Given: plane đ?›ź, lines đ?‘‘1 , đ?‘‘2 and points đ??´, đ??ľ ∉ đ?›ź âˆŞ đ?‘‘1 âˆŞ đ?‘‘2 . Find a point đ?‘€ ∈ đ?›ź such that the lines đ?‘€đ??´, đ?‘€đ??ľ intersect đ?‘‘1 and đ?‘‘2 . Solution to Problem 130

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131. There are given the plane đ?›ź, the line đ?‘‘ ∉ đ?›ź, the points đ??´, đ??ľ ∉ đ?›ź âˆŞ đ?‘‘, and đ??ś ∈ đ?›ź. Let đ?‘€ ∈ đ?‘‘ and đ??´â€˛, đ??ľâ€˛ the points of intersection of the lines đ?‘€đ??´, đ?‘€đ??ľ with plane đ?›ź (if they exist). Determine the point đ?‘€ such that the points đ??ś, đ??´â€˛, đ??ľâ€˛ to be collinear. Solution to Problem 131

132. If points đ??´ and đ??ľ of an open half-space đ?œŽ, then [đ??´đ??ľ] ⊂ đ?œŽ. The property is as well adherent for a closed half-space. Solution to Problem 132

133. If point đ??´ is not situated on plane đ?›ź and đ??ľ ∈ đ?›ź then |đ??ľđ??´ ⊂ |đ?›źđ??´. Solution to Problem 133

134. Show that the intersection of a line đ?‘‘ with a half-space is either line đ?‘‘ or a ray or an empty set. Solution to Problem 134

135. Show that if a plane đ?›ź and the margin of a half-space đ?œŽ are secant planes, then the intersection đ?œŽ ∊ đ?›ź is a half-plane. Solution to Problem 135

136. The intersection of a plane � with a half-space is either the plane � or a half-plane, or an empty set. Solution to Problem 136

137. Let đ??´, đ??ľ, đ??ś, đ??ˇ four non coplanar points and đ?›ź a plane that doesn’t pass through one of the given points, but it passes trough a point of the line |đ??´đ??ľ|. How many of the segments |đ??´đ??ľ|, |đ??´đ??ś|, |đ??´đ??ˇ|, |đ??ľđ??ś|, |đ??ľđ??ˇ|, |đ??śđ??ˇ| can be intersected by plane đ?›ź? Solution to Problem 137

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255 Compiled and Solved Problems in Geometry and Trigonometry

138. Let đ?‘‘ be a line and đ?›ź, đ?›˝ two planes such that đ?‘‘ ∊ đ?›˝ = ∅ and đ?›ź ∊ đ?›˝ = ∅. Show that if đ??´ ∈ đ?‘‘ and đ??ľ ∈ đ?›ź, then đ?‘‘ ⊂ |đ?›˝đ??´ and đ?›ź ⊂ |đ?›˝đ??ľ. Solution to Problem 138

139. Let |đ?›źđ??´ and |đ?›˝đ??ľ be two half-spaces such that đ?›ź ≠đ?›˝ and |đ?›źđ??´ ⊂ |đ?›˝đ??ľ or |đ?›źđ??´ ∊ |đ?›˝đ??ľ = ∅. Show that đ?›ź ∊ đ?›˝ = ∅. Solution to Problem 139

140. Show that the intersection of a dihedral angle with a plane � can be: a right angle, the union of two lines, a line, an empty set or a closed halfplane and cannot be any other type of set. Solution to Problem 140

141. Let đ?‘‘ be the edge of a proper dihedron ∠đ?›ź ′ đ?›˝ ′ , đ??´ ∈ đ?›ź ′ – đ?‘‘, đ?‘? ∈ đ?›˝ ′ – đ?‘‘ and đ?‘ƒ ∈ int. ∠đ?›ź ′ đ?›˝ ′ . Show that: a. (đ?‘ƒđ?‘‘) ∊ int. ∠đ?›ź ′ đ?›˝ ′ = |đ?‘‘đ?‘ƒ; b. If đ?‘€ ∈ đ?‘‘, int. ∠đ??´đ?‘€đ??ľ = int. đ?›ź ′ đ?›˝ ′ ∊ (đ??´đ?‘€đ??ľ). Solution to Problem 141

142. Consider the notes from the previous problem. Show that: a. The points đ??´ and đ??ľ are on different sides of the plane (đ?‘ƒđ?‘‘); b. The segment |đ??´đ??ľ| and the half-plane |đ?‘‘đ?‘ƒ have a common point. Solution to Problem 142

143. If ∠đ?‘Žđ?‘?đ?‘? is a trihedral angle, đ?‘ƒ ∈ int. ∠đ?‘Žđ?‘?đ?‘? and đ??´, đ??ľ, đ??ś are points on edges đ?‘Ž, đ?‘?, đ?‘?, different from đ?‘‚, then the ray |đ?‘‚đ?‘ƒ and int. đ??´đ??ľđ??ś have a common point. Solution to Problem 143

144. Show that any intersection of convex sets is a convex set. Solution to Problem 144 111

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145. Show that the following sets are convex planes, half-planes, any open or closed half-space and the interior of a dihedral angle. Solution to Problem 145

146. Can a dihedral angle be a convex set? Solution to Problem 146

147. Which of the following sets are convex: a. a trihedral angle; b. its interior; c. the union of its faces; d. the union of its interior with all its faces? Solution to Problem 147

148. Let đ?œŽ be an open half-space bordered by plane đ?›ź and đ?‘€ a closed convex set in plane đ?›ź. Show that the set đ?‘€ ∊ đ?œŽ is convex. Solution to Problem 148

149. Show that the intersection of sphere đ?‘†(đ?‘‚, đ?‘&#x;) with a plane which passes through đ?‘‚, is a circle. Solution to Problem 149

150. Prove that the int. đ?‘†(đ?‘‚, đ?‘&#x;) is a convex set. Solution to Problem 150

151. Show that, by unifying the midpoints of the opposite edges of a tetrahedron, we obtain concurrent lines. Solution to Problem 151

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255 Compiled and Solved Problems in Geometry and Trigonometry

152. Show that the lines connecting the vertices of a tetrahedron with the centroids of the opposite sides are concurrent in the same point as the three lines from the previous example. Solution to Problem 152

153. Let đ??´đ??ľđ??śđ??ˇ be a tetrahedron. We consider the trihedral angles which have as edges [đ??´đ??ľ, [đ??´đ??¸, [đ??´đ??ˇ, [đ??ľđ??´, [đ??ľđ??ś, [đ??ľđ??ˇ, [đ??śđ??´, [đ??śđ??ľ, [đ??śđ??ˇ, [đ??ˇđ??´, [đ??ˇđ??ľ, [đ??ˇđ??ś. Show that the intersection of the interiors of these 4 trihedral angles coincides with the interior of tetrahedron [đ??´đ??ľđ??śđ??ˇ]. Solution to Problem 153

154. Show that (∀) đ?‘€ ∈ int. [đ??´đ??ľđ??śđ??ˇ] (∃) đ?‘ƒ ∈ |đ??´đ??ľ| and đ?‘„ ∈ |đ??śđ??ˇ| such that đ?‘€ ∈ ‖đ?‘ƒđ?‘„. Solution to Problem 154

155. The interior of tetrahedron [đ??´đ??ľđ??śđ??ˇ] coincides with the union of segments |đ?‘ƒđ?‘„| with đ?‘ƒ ∈ |đ??´đ??ľ| and đ?‘„ ∈ |đ??śđ??ˇ|, and tetrahedron [đ??´đ??ľđ??śđ??ˇ] is equal to the union of the closed segments [đ?‘ƒđ?‘„], when đ?‘ƒ ∈ [đ??´đ??ľ] and đ?‘„ ∈ [đ??śđ??ˇ]. Solution to Problem 155

156. The tetrahedron is a convex set. Solution to Problem 156

157. Let đ?‘€1 and đ?‘€2 convex sets. Show that by connecting segments [đ?‘ƒđ?‘„], for which đ?‘ƒ ∈ đ?‘€1 and đ?‘„ ∈ đ?‘€2 we obtain a convex set. Solution to Problem 157

158. Show that the interior of a tetrahedron coincides with the intersection of the open half-spaces determined by the planes of the faces and the opposite peak. Define the tetrahedron as an intersection of half-spaces. Solution to Problem 158

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Solutions Solution to Problem 118. We assume that đ?‘‘ ∊ đ?›ź = {đ??´, đ??ľ} ⇒ đ?‘‘ ⊂ đ?›ź. It contradicts the hypothesis â&#x;š đ?‘‘ ∊ đ?›ź = {đ??´} or đ?‘‘ ∊ đ?›ź = ∅.

Solution to Problem 119. We assume that all the points belong to the plane đ?›ź â&#x;š (∄) for the points that are not situated in the same plane. False!

Solution to Problem 120. ∃ đ??´, đ??ľ, đ??ś, đ??ˇ, which are not in the same plane. We assume that đ??´đ??ľ ∊ đ??śđ??ˇ = {0} â&#x;š đ??´đ??ľ and đ??śđ??ˇ are contained in the same plane and thus đ??´, đ??ľ, đ??ś, đ??ˇ are in the same plane. False, it contradicts the hypothesis â&#x;š đ??´đ??ľ ∊ đ??śđ??ˇ = ∅ â&#x;š (∃)

lines with no

point in common.

Solution to Problem 121. (∃) đ??´ ∉ đ?‘‘ (if all the points would ∈ đ?‘‘, the existence of the plane and space would be negated). Let đ?›ź = (đ?‘‘đ??´), (∃)đ??ľ ∉ đ?›ź (otherwise the space wouldn’t exist). Let đ?›˝ = (đ??ľđ?‘‘), đ?›ź ≠đ?›˝ and both contain line đ?‘‘.

Solution to Problem 122. We show that � ≠�′ ≠�′′ ≠�. Let

đ?‘‘⊂đ?›ź đ?‘‘ ∊ đ?‘‘′ = {đ??´} = (đ?‘‘, đ?‘‘′ ) â&#x;š { ′ đ?‘‘ ⊂đ?›ź đ?‘‘ ∊ đ?‘‘′ = {đ??ľ} â&#x;š đ??ľ ∈ đ?‘‘ â&#x;š đ??ľ ∈ đ?‘Ž , đ??ľ ∈ đ?‘‘′ { đ?‘‘ ⊂âˆ? đ??ľâ‰ đ??´

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đ?‘‘′′ ∊ đ?‘‘′ = {đ??ś} ′ â&#x;š đ??śâ€˛ ∈ đ?‘‘ â&#x;š đ??ś ∈ đ?›ź, đ??ś ∈ đ?‘‘′′ đ??śâ‰ đ??ľ đ?‘‘ ⊂âˆ? đ??śâ‰ đ??´ đ?‘‘ = đ?‘‘′ â&#x;š or đ?‘‘ = đ?‘‘′′ â&#x;š đ?‘‘′′ ⊂ đ?›ź, so the lines are located on the same plane Îą.

If

đ?‘‘ ∊ đ?‘‘′ = {đ??´} â&#x;š đ??´ ∈ đ?‘‘′ } â&#x;š đ?‘‘′ ∊ đ?‘‘′′ = {đ??´}, and the three lines have a point in đ?‘‘′′ ∊ đ?‘‘ = {đ??´} â&#x;š đ??´ ∈ đ?‘‘′′

common.

Solution to Problem 123.

a. đ??ˇ ∉ (đ??´đ??ľđ??ś). We assume that đ??ˇ, đ??´, đ??ľ collinear â&#x;š (∃)đ?‘‘ such that

đ??ˇ ∈ đ?‘‘, đ??´ ∈ đ?‘‘, đ??ľ ∈ đ?‘‘ đ?‘‡2 }⇒ đ?‘‘ ⊂ đ??´ ∈ (đ??´đ??ľđ??ś), đ??ľ ∈ (đ??´đ??ľđ??ś)

(đ??´đ??ľđ??ś) â&#x;š đ??ˇ ∈ (đ??´đ??ľđ??ś) – false. Therefore, the points đ??ˇ, đ??´, đ??ľ are not collinear. b. Let (đ??ˇđ??´đ??ľ) ∊ (đ??ľđ??śđ??ˇ) ∊ (đ??ˇđ??śđ??´) = đ??¸. As the planes are distinct, their intersections are:

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(đ??ˇđ??´đ??ľ) ∊ (đ??ˇđ??ľđ??ś) = đ??ˇđ??ľ â&#x;š If (đ??ˇđ??´đ??ľ) = (đ??ˇđ??ľđ??ś) (đ??ˇđ??´đ??ľ) ∊ (đ??ˇđ??śđ??´) = đ??ˇđ??´ } â&#x;š đ??´, đ??ľ, đ??ś, đ??ˇ coplanar, contrary to the hypothesis. (đ??ˇđ??ľđ??ś) ∊ (đ??ˇđ??śđ??´) = đ??ˇđ??ś đ?‘€ ∈ đ??ˇđ??ľ đ??ľ ∈ đ?‘€đ??ˇ We suppose that (∃)đ?‘€ ∈ đ??¸, đ?‘€ ≠đ??ˇ â&#x;š }â&#x;š } â&#x;š đ??´, đ??ľ, đ??ˇ are collinear đ?‘€ ∈ đ??ˇđ??´ đ??´ ∈ đ?‘€đ??ˇ (false, contrary to point a.). Therefore, set đ??¸ has a single point đ??¸ = {đ??ˇ}.

Solution to Problem 124.

We showed at the previous exercise that if đ??ˇ ∉ (đ??´đ??ľđ??ś), (đ??ˇđ??´đ??ľ) ≠(đ??ˇđ??ľđ??ś). We show that đ??¸, đ??š, đ??ş are not collinear. We assume the opposite. Then,

Having three common points đ??ˇ, đ??ľ and đ??ş â&#x;š false. So đ??¸, đ??š, đ??ş are not collinear and determine a plane (đ??¸đ??šđ??ş).

â&#x;š đ?‘ƒ, đ?‘„, đ?‘… are collinear because ∈ to the line of intersection of the two planes. 116

255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 125. The planes are (đ??´, đ?‘‘′); (đ??ľ, đ?‘‘′); (đ??ś, đ?‘‘′).

Generalization: The number of planes corresponds to the number of points on line � because �′ contains only 2 points.

Solution to Problem 126. Let line đ?‘‘ be given, and đ??´ any point such that đ??´ ∉ đ?‘‘.

We obtain the plane đ?›ź = (đ??´, đ?‘‘), and let đ?‘€ ∉ đ?›ź. The line đ?‘‘′ = đ??´đ?‘€, đ?‘‘′ ⊄ đ?›ź is not thus contained in the same plane with đ?‘‘. The desired planes are those of type (đ?‘€đ?‘‘), đ?‘€ ∈ đ?‘‘′, that is an infinity of planes.

Solution to Problem 127. a. (∀) 3 points determine a plane. Let plane (đ??´đ??ľđ??ˇ). We choose in this plane đ?‘ƒ ∈ |đ??´đ??ˇ| and đ?‘„ ∈ |đ??´đ??ľ| such that đ?‘ƒ ∈ |đ??ľđ?‘„|, then the line đ?‘ƒđ?‘„ separates the points đ??´ and đ??ˇ, but does not separate đ??´ and đ??ľ, so it separates đ?‘ƒ and đ??ˇ ⇒ đ?‘ƒđ?‘„ ∊ |đ??ľđ??ˇ| = đ?‘…, where đ?‘… ∈ |đ??ľđ??ˇ|. Thus, the line đ?‘ƒđ?‘„ meets 3 of the given lines. Let’s see if it can meet more. 117

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We assume that

it has two points in common with the plane.

â&#x;š đ??´, đ??ľ, đ??ś, đ??ˇ coplanar – false. Thus,

false. We show in the same way that đ?‘ƒđ?‘„ does not cut đ??´đ??ś or đ??ˇđ??ś, so a line meets at most three of the given lines. b. We consider points đ??¸, đ??š, đ??ş such that đ??¸ ∈ |đ??ľđ??ś|, đ??´ ∈ |đ??ˇđ??š|, đ??ˇ ∈ |đ??ľđ??ş|. These points determine plane (đ??¸đ??šđ??ş) which obviously cuts the lines đ??ľđ??ś, đ??ľđ??ˇ and đ??ľđ??ˇ. đ??šđ??ş does not separate đ??´ and đ??ˇ or đ??ľđ??ˇ â&#x;š it does not separate đ??´ or đ??ľ â&#x;š đ??´ ∈ |đ??ľđ?‘…|.

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Let’s show that (đ??¸đ??šđ??ş) meets as well the lines đ??´đ??ľ, đ??śđ??ˇ, đ??´đ??ś. In the plane (đ??´đ??ľđ??ˇ) we consider the triangle đ??šđ??ˇđ??ş and the line đ??´đ??ľ. As this line cuts side |đ??šđ??ˇ|, but it does not cut |đ??ˇđ??ş|, it must cut side |đ??šđ??ş|, so đ??´đ??ľ ∊ |đ??šđ??ş| = {đ?‘…} â&#x;š đ?‘… ∈ |đ??šđ??ş| ⊂ (đ??¸đ??šđ??ş), so đ??´đ??ľ ∊ (đ??¸đ??šđ??ş) = {đ?‘…}. In the plane (đ??ľđ??śđ??ˇ), the line đ??¸đ??ş cuts |đ??ľđ??ś| and does not cut |đ??ľđ??ˇ|, so đ??¸đ??ş cuts the side |CD|, đ??¸đ??ş ∊ |đ??śđ??ˇ| = {đ?‘ƒ} â&#x;š đ?‘ƒ ∈ đ??¸đ??ş ⊂ (đ??¸đ??šđ??ş) â&#x;š đ??śđ??ˇ ∊ (đ??¸đ??šđ??ş) = {đ?‘ƒ}. đ?‘… ∈ (đ??¸đ??šđ??ş), đ?‘… does not separate đ??´ and đ??ľ } â&#x;š đ?‘… ∈∊ |đ??´đ??ś| = đ?‘„ â&#x;š đ?‘„ ∈ đ?‘…đ??¸ â&#x;š đ?‘„ đ??¸ separates đ??ľ and đ??ś ∈ (đ??¸đ??šđ??ş) ∊ đ??´đ??ś = {đ?‘„}.

Solution to Problem 128. We assume problem is solved, if

đ?‘€âˆˆđ?›ź } â&#x;š đ?›ź ∊ đ?›˝ ≠∅, â&#x;š đ?›ź ∊ đ?›˝ = đ?‘‘. đ?‘€âˆˆđ?›˝

As ||đ?‘€đ??´|| = ||đ?‘€đ??ľ|| â&#x;š đ?‘€ ∈ the bisecting line of the segment [đ??´đ??ľ]. So, to find đ?‘€, we proceed as follows: 1. We look for the line of intersection of planes đ?›ź and đ?›˝, d. If đ?›ź âˆĽ đ?›˝, the problem hasn’t got any solution. 2. We construct the bisecting line đ?‘‘′ of the segment [đ??´đ??ľ] in the plane đ?›ź. 3. We look for the point of intersection of lines đ?‘‘ and đ?‘‘′. If đ?‘‘ âˆĽ đ?‘‘′, the problem hasn’t got any solution.

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Solution to Problem 129. If đ?›ź ∊ đ?›˝ = ∅ â&#x;š đ?›ź ∊ đ?›˝ ∊ đ?›ž = ∅. If đ?›ź ∊ đ?›˝ = đ?‘‘, the desired intersection is đ?‘‘ ∊ đ?›ž, which can be a point (the 3 planes are concurrent), the empty set (the line of intersection of two planes is || with the third) or line đ?‘‘ (the 3 planes which pass through đ?‘‘ are secant).

Solution to Problem 130. To determine đ?‘€, we proceed as follows: 1. We construct plane (đ??´đ?‘‘1 ) and we look for the line of intersection with đ?›ź1 , đ?‘‘1 . If đ?‘‘1 (/∃), ∄ neither does đ?‘€. 2. We construct plane (đ??ľđ?‘‘2 ) and we look for the line of intersection with đ?›ź, đ?‘‘2 ′. If đ?‘‘2 ′ does not exist, neither does đ?‘€. 3. We look for the point of intersection of lines đ?‘‘1 ′ and đ?‘‘2 ′. The problem has only one solution if the lines are concurrent, an infinity if they are coinciding lines and no solution if they are parallel.

Solution to Problem 131.

We assume the problem is solved. a. First we assume that đ??´. đ??ľ, đ??ś are collinear. As đ??´đ??´â€˛ and đ??ľđ??ľâ€˛ are concurrent lines, they determine a plane đ?›˝, that intersects đ?›ź after line đ??´â€˛đ??ľâ€˛. As

and points đ??ś, đ??´, đ??ľâ€˛ are collinear (∀)đ?‘€ ∈ đ?‘‘.

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b. We assume that đ??´, đ??ľ, đ??ś are not collinear. We notice that: (đ??´đ??´â€˛, đ??ľđ??ľâ€˛) = đ?›˝ (plane determined by 2 concurrent lines). đ?›˝ âˆŞ đ?›ź = đ?‘‘′ and đ??ś ∈ đ?‘‘′.

To determine đ?‘€ we proceed as follows: 1)

We determine plane (đ??´đ??ľđ??ś);

2)

We look for the point of intersection of this plane with line đ?‘‘, so đ?‘‘ ∊

(đ??´đ??ľđ??ś) = {đ?‘€} is the desired point. Then (đ??´đ??ľđ??ś) ∊ đ?›ź = đ?‘‘′ .

â&#x;š đ??´â€˛, đ??ľâ€˛, đ??śâ€˛ are collinear.

Solution to Problem 132.

đ??´ ∈ đ?œŽ and đ??ľ ∈ đ?œŽ â&#x;š [đ??´đ??ľ] ∊ đ?›ź ≠∅. Let đ?œŽ = |đ?›źđ??´ = |đ?›źđ??ľ. Let đ?‘€ ∈ |đ??´đ??ľ| and we must show that đ?‘€ ∈ đ?œŽ(∀)đ?‘€ inside the segment. We assume the contrary that đ?‘€ ∉ đ?œŽ â&#x;š (∃)đ?‘ƒ such that [đ??´đ?‘€] ∊ đ?‘‘ = {đ?‘ƒ} â&#x;š đ?‘ƒ ∈ [đ??´đ?‘€] â&#x;š đ?‘ƒ ∈ [đ??´đ??ľ] â&#x;š [đ??´đ??ľ] ∊ đ?›ź ≠∅ false. đ?‘ƒ ∈ đ?›ź, so đ?‘€ ∈ đ?œŽ. The property is also maintained for the closed half-space.

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Compared to the previous case there can appear the situation when one of the points đ??´ and đ??ľ ∈ đ?›ź or when both belong to đ?›ź.

If đ??´ ∈ đ?›ź, đ??ľ ∈ đ?œŽ, |đ??´đ??ľ| ∊ đ?›ź ≠∅ and we show as we did above that:

If:

Solution to Problem 133.

Let

So

Solution to Problem 134. Let đ?›ź be a plane and đ?œŽ1 , đ?œŽ2 the two half-spaces that it determines. We consider half-space đ?œŽ1 .

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255 Compiled and Solved Problems in Geometry and Trigonometry

đ?‘ƒ determines on đ?‘‘ two rays, |đ?‘ƒđ??´ and |đ?‘ƒđ??ľ where

đ??´ and đ??ľ are in different half-spaces. We assume

Solution to Problem 135.

Let đ?œŽ be an open half-space and đ?‘? its margin and let đ?‘‘ = đ?›ź ∊ đ?›˝. We choose points đ??´ and đ??ľ ∈ đ?›ź – đ?‘‘, on both sides of line đ?‘‘ â&#x;š

â&#x;š đ??´, đ??ľ are on one side and on the other side of đ?›˝ and it means that only one of them is on đ?œŽ. We assume that đ??´ ∈ đ?œŽ â&#x;š đ??ľ ∈ đ?œŽ. We now prove đ?›ź ∊ đ?œŽ = |đ?‘‘đ??´. đ?›ź ∊ đ?œŽ ⊂ |đ?‘‘đ??´ 123

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Let

[đ?‘€đ??ľ] ∈ đ?‘‘ ≠∅ â&#x;š đ?‘€ and đ??ľ are on one side and on the other side of line đ?‘‘ â&#x;š đ?‘€ is on the same side of line đ?‘‘ with đ??´ â&#x;š đ?‘€ ∈ |đ?‘‘đ??´

â&#x;š đ?‘€ ∈ đ?‘Ž ∊ đ?œŽ, so |đ?‘‘đ??´ ⊂ đ?‘Ž ∊ đ?œŽ.

Solution to Problem 136. Let Ďƒ be the considered half-space and β its margin. There are more possible cases:

In this case it is possible that:

Let

is a half-plane according to a previous problem.

Solution to Problem 137.

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255 Compiled and Solved Problems in Geometry and Trigonometry

The intersection of two planes is a line and it cuts only two sides of a triangle. There are more possible cases: 1. đ?‘‘ cuts |đ??´đ??ľ| and |đ??ľđ??ś| đ?‘‘′ cuts |đ??´đ??ľ| and |đ??´đ??ˇ|, đ?›ź cuts |đ??´đ??ˇ| so it has a point in common with (đ??´đ??ˇđ??ś) and let (đ??´đ??ˇđ??ś) ∊ đ?›ź = đ?‘‘′′. đ?‘‘′′ cuts |đ??´đ??ˇ| and does not cut |đ??´đ??ś| â&#x;š đ?‘‘′′ cuts |đ??ˇđ??ś| đ?›ź cuts |đ??ˇđ??ś| and |đ??ľđ??ś| â&#x;š it does not cut |đ??ľđ??ˇ|. In this case đ?›ź cuts 4 of the 6 segments (the underlined ones). 2. đ?‘‘ cuts |đ??´đ??ľ| and |đ??´đ??ś|, it does not cut |đ??ľđ??ˇ| đ?‘‘′ cuts |đ??´đ??ľ| an |đ??´đ??ˇ|, it does not cut |đ??ľđ??ˇ| đ?‘‘′′ cuts |đ??´đ??ˇ| an |đ??´đ??ś|, it does not cut |đ??ˇđ??ś| â&#x;š đ?›ź does not intersect plane (đ??ľđ??śđ??ˇ). In this case đ?›ź intersects only 3 of the 6 segments. 3. đ?‘‘ cuts |đ??´đ??ľ| and |đ??ľđ??ś|, it does not cut |đ??´đ??ś| đ?‘‘′ cuts |đ??´đ??ľ| an |đ??ľđ??ˇ|, it does not cut |đ??ˇđ??ś| đ?›ź intersects |đ??ľđ??ˇ| and |đ??ľđ??ś|, so it does not cut |đ??ˇđ??ś| In ∆đ??ľđ??ˇđ??ś â&#x;š đ?›ź does not intersect plane (đ??´đ??ˇđ??ś) In this case đ?›ź intersects only three segments. 4. đ?‘‘ cuts |đ??´đ??ľ| and |đ??´đ??ś|, it does not cut |đ??ľđ??ś| đ?‘‘′ cuts |đ??´đ??ľ| an |đ??ľđ??ˇ|, it does not cut |đ??´đ??ˇ| đ?‘‘′′ cuts |đ??´đ??ś| an |đ??ˇđ??ś| đ?›ź does not cut |đ??ľđ??ś| in triangle đ??ľđ??ˇđ??ś. So đ?›ź intersects 4 or 3 segments.

Solution to Problem 138. Let

Let

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Solution to Problem 139. We first assume that đ?›ź ≠đ?›˝ and |đ?›źđ??´ ⊂ |đ?›˝đ??ľ.

As

The hypothesis can then be written as đ?›ź ≠đ?›˝ and |đ?›źđ??´ ⊂ |đ?›˝đ??ľ. Let’s show that đ?›ź ∊ đ?›˝ = ∅. By reductio ad absurdum, we assume that đ?›ź ∊ đ?›˝ ≠∅ â&#x;š (∃)đ?‘‘ = đ?›ź ∊ đ?›˝ and let đ?‘‚ ∈ đ?‘‘, so đ?‘‚ ∈ đ?›ź and đ?‘‚ ∈ đ?›˝. We draw through đ??´ and đ?‘‚ a plane đ?‘&#x;, such that đ?‘‘ ∈ đ?‘&#x;, so the three planes đ?›ź, đ?›˝ and đ?‘&#x; do not pass through this line. As đ?‘&#x; has the common point đ?‘‚ with đ?›ź and đ?›˝, it is going to intersect these planes.

which is a common point of the 3 planes. Lines đ?›ż and đ?›żâ€˛ determine 4 angles in plane đ?‘&#x;, having đ?‘‚ as a common peak, đ??´ ∈ the interior of one of them, let đ??´ ∈ int. Ě‚ . We consider đ??ś ∈ int. â„Žđ?‘˜ Ě‚. â„Žđ?‘˜ Then đ??ś is on the same side with đ??´ in relation to đ?›żâ€˛, so đ??ś is on the same side with đ??´ in relation to đ?›ź â&#x;š đ??ś ∈ |đ?›źđ??´.

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But đ??ś is on the opposite side of đ??´ in relation to đ?›ż, so đ??ś is on the opposite side of đ??´ in relation to đ?›˝ â&#x;š đ??ś ∉ |đ?›˝đ??´. So |đ?›źđ??´ ⊄ |đ?›˝đ??´ – false – it contradicts the hypothesis â&#x;š So đ?›ź ∊ đ?›˝ = ∅.

Solution to Problem 140. Let đ?‘‘ be the edge of the given dihedral angle. Depending on the position of a line in relation to a plane, there can be identified the following situations:

̂ = �′�′′ ̂ thus an angle. The ray with its origin in �, so � ⊂ �′�′

Indeed, if we assumed that đ?‘‘′ ∊ đ?‘‘′′ ≠∅ â&#x;š (∃)đ?‘‚ ∈ đ?‘‘′ ∊ đ?‘‘′′.

false – it contradicts the hypothesis.

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Or

′ đ?›ž ′ = đ?‘‘′′ - a line. Ě‚ in this case đ?›ź ∊ đ?›˝

′ đ?›ž ′ = ∅. Ě‚ Then đ?›ź ∊ đ?›˝

đ?‘‘ ∊ đ?›ź = đ?‘‘, but đ?›ź ≠đ?›˝, đ?›ź ≠đ?›ž ′ đ?›ž ′ = đ?‘‘ thus the intersection is a line. Ě‚ đ?›źâˆŠđ?›˝

In this case the intersection is a closed half-plane.

Solution to Problem 141.

is a half-plane

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255 Compiled and Solved Problems in Geometry and Trigonometry

is a half-plane

From (*) and (**),

so they are secant planes

Solution to Problem 142.

â&#x;š points đ??´ and đ??ľ are on different sides of (đ?‘‘đ?‘ƒ).

Solution to Problem 143.

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Let rays As đ?‘ƒ is interior to the dihedron formed by any half-plane passing through đ?‘‚ of the trihedral, so

So

đ?‘ƒ and đ?‘„ in the same half-plane det. đ?‘‚đ??´ ⇒ đ?‘ƒ and đ?‘„ on the same side of đ?‘‚đ??´ (1) â&#x;š đ?‘ƒ and đ??´ are on the same side of (đ?‘‚đ??ľđ??ś) ∊ đ?›žâ€˛ â&#x;š đ?‘ƒ and đ??´ are on the same side of đ?‘‚đ?‘„ (2). From (1) and (2) â&#x;š

Solution to Problem 144. Let đ?‘€ and đ?‘€â€˛â€˛be two convex sets and đ?‘€ ∊ đ?‘€â€˛ their intersection. Let

so the intersection is convex.

Solution to Problem 145.

a. Let đ?‘ƒ, đ?‘„ ∈ đ?›ź; đ?‘ƒ ≠đ?‘„ â&#x;š |đ?‘ƒđ?‘„ = đ?‘ƒđ?‘„ (the line is a convex set) đ?‘ƒđ?‘„ ⊂ đ?›ź, so |đ?‘ƒđ?‘„| ⊂ đ?›ź, so the plane is a convex set. 130

255 Compiled and Solved Problems in Geometry and Trigonometry

b. Half-planes: Let đ?‘† = |đ?‘‘đ??´ and đ?‘ƒ, đ?‘„ ∈ đ?‘† â&#x;š |đ?‘ƒđ?‘„| ∊ đ?‘‘ = ∅. Let đ?‘€ ∈ |đ?‘ƒđ?‘„| â&#x;š |đ?‘ƒđ?‘€| ⊂ |đ?‘ƒđ?‘„| â&#x;š |đ?‘ƒđ?‘€| ∊ đ?‘‘ = ∅ â&#x;š đ?‘ƒ and đ?‘€ are in the same half-plane â&#x;š đ?‘€ ∈ đ?‘†. So |đ?‘ƒđ?‘„| ⊂ đ?‘† and đ?‘† is a convex set.

Let đ?‘†â€˛ = [đ?‘‘đ??´. There are three situations: 1)

đ?‘ƒ, đ?‘„ ∈ |đ?‘‘đ??´ – previously discussed;

2)

đ?‘ƒ, đ?‘„ ∈ đ?‘‘ â&#x;š |đ?‘ƒđ?‘„| ⊂ đ?‘‘ ⊂ đ?‘†â€˛;

3)

đ?‘ƒ ∈ đ?‘‘, đ?‘„ ∉ đ?‘‘ â&#x;š |đ?‘ƒđ?‘„| ⊂ |đ?‘‘đ?‘„ â&#x;š |đ?‘ƒđ?‘„| ⊂ |đ?‘‘đ??´ ⊂ [đ?‘‘đ??´ so [đ?‘‘đ??´ is a convex set.

c. Half-spaces: Let đ?œŽ = |đ?›źđ??´ and let đ?‘ƒ, đ?‘„ ∈ đ?œŽ â&#x;š |đ?‘ƒđ?‘„| ∊ đ?›ź = ∅. Let đ?‘€ ∈ |đ?‘ƒđ?‘„| â&#x;š |đ?‘ƒđ?‘€| ⊂ |đ?‘ƒđ?‘„| â&#x;š |đ?‘ƒđ?‘€| ∊ đ?›ź = ∅. Let đ?œŽâ€˛ = [đ?›źđ??´. There are three situations: 1)

đ?‘ƒ, đ?‘„ ∈ |đ?›źđ??´ previously discussed;

2)

đ?‘ƒ, đ?‘„ ∈ đ?›ź â&#x;š |đ?‘ƒđ?‘„| ⊂ đ?›ź ⊂ đ?œŽâ€˛;

3)

đ?‘ƒ ∈ đ?›ź, đ?‘„ ∉ đ?›ź.

and so đ?œŽâ€˛ is a convex set.

d. the interior of a dihedral angle: đ?‘–đ?‘›đ?‘Ą. đ?›źâ€˛đ?›˝â€˛ = |đ?›źđ??´ ∊ |đ?›˝đ??ľ and as each half-space is a convex set and their intersection is the convex set.

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Solution to Problem 146. No. The dihedral angle is not a convex set, because if we consider it as in the previous figure đ??´ ∈ đ?›˝â€˛ and đ??ľ ∈ đ?›źâ€˛.

Only in the case of the null or straight angle, when the dihedral angle becomes a plane or closed half-plane, is a convex set.

Solution to Problem 147.

a. No. The trihedral angle is not the convex set, because, if we take đ??´ ∈ đ?‘Ž and đ?‘„ ∈ Ě‚ determined by đ?‘ƒ ∈ the int. đ?‘Žđ?‘?đ?‘? Ě‚ , (∃)đ?‘… such that |đ?‘‚đ?‘ƒ ∊ the int. đ??´đ??ľđ??ś = the int. đ?‘?đ?‘? Ě‚. {đ?‘…}, đ?‘… ∈ |đ??´đ?‘„|, đ?‘… ∉ đ?‘Žđ?‘?đ?‘? Ě‚ , but |đ??´đ?‘„| ∉ đ?‘Žđ?‘?đ?‘?. So đ??´, đ?‘„ ∈ đ?‘Žđ?‘?đ?‘? b. đ?›Ł = (đ?‘‚đ??śđ??´), đ?›ž = (đ?‘‚đ??´đ??ľ) is a convex set as an intersection of convex sets. C) It is the same set from a. and it is not convex. D) The respective set is [đ?›źđ??´ ∊ [đ?›˝đ??ľ ∊ [đ?›žđ??ś, intersection of convex sets and, thus, it is convex.

Solution to Problem 148.

Let đ?œŽ = |đ?›źđ??´ and đ?‘€ ⊂ đ?›ź. Let đ?‘ƒ, đ?‘„ ∈ đ?‘€ ∊ đ?œŽ. We have the following situations: 132

255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 149.

Solution to Problem 150.

Let

In plane (đ?‘‚đ?‘ƒđ?‘„), let đ?‘€ ∈ (đ?‘ƒđ?‘„). or

Solution to Problem 151. Let:

đ?‘ƒ midpoint of |đ??´đ??ľ| đ?‘… midpoint of |đ??ľđ??ś| đ?‘„ midpoint of |đ??ˇđ??ś| đ?‘† midpoint of |đ??´đ??ˇ| đ?‘‡ midpoint of |đ??ľđ??ˇ| đ?‘ˆ midpoint of |đ??´đ??ś|

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In triangle ABC:

In triangle DAC:

â&#x;š parallelogram â&#x;š |đ?‘ƒđ?‘„| and |đ?‘†đ?‘…| intersect at their midpoint đ?‘‚.

â&#x;š đ?‘†đ?‘‡đ?‘…đ?‘ˆ parallelogram. â&#x;š |đ?‘‡đ?‘ˆ| passes through midpoint đ?‘‚ of |đ?‘†đ?‘…|. Thus the three lines đ?‘ƒđ?‘…, đ?‘†đ?‘…, đ?‘‡đ?‘ˆ are concurrent in đ?‘‚.

Solution to Problem 152.

Let tetrahedron đ??´đ??ľđ??śđ??ˇ and đ??¸ be the midpoint of |đ??śđ??ˇ|. The centroid đ??ş of the face đ??´đ??śđ??ˇ is on |đ??´đ??¸| at a third from the base. The centroid đ??şâ€˛ of the face đ??ľđ??śđ??ˇ is on |đ??ľđ??¸| at a third from the base |đ??śđ??ˇ|. We separately consider ∆đ??´đ??¸đ??ľ. Let đ??š be the midpoint of đ??´đ??ľ, so đ??¸đ??š is median in this triangle and, in the previous problem, it was one of the 3 concurrent segments in a point located in the middle of each. Let đ?‘‚ be the midpoint of |đ??¸đ??š|. We write đ??´đ?‘‚ ∊ đ??¸đ??ľ = {đ??şâ€˛} and đ??ľđ??ş ∊ đ??¸đ??´ = {đ??ş}.

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255 Compiled and Solved Problems in Geometry and Trigonometry

From (1) and (2)

â&#x;š đ??şâ€˛ is exactly the centroid of face đ??ľđ??śđ??ˇ, because it is situated on median |đ??¸đ??ľ| at a third from đ??¸. We show in the same way that đ??ş is exactly the centroid of face đ??´đ??śđ??ˇ. We’ve thus shown that đ??ľđ??ş and đ??´đ??şâ€˛ pass through point đ?‘‚ from the previous problem. We choose faces đ??´đ??śđ??ˇ and đ??´đ??śđ??ľ and mark by đ??şâ€˛â€˛ the centroid of face đ??´đ??śđ??ľ, we show in the same way that đ??ľđ??ş and đ??ˇđ??şâ€˛â€˛ pass through the middle of the segment |đ?‘€đ?‘ | (|đ??´đ?‘€| ≥ |đ?‘€đ??ś|, |đ??ľđ?‘ | ≥ |đ?‘ đ??ˇ|) thus also through point đ?‘‚, etc.

Solution to Problem 153.

We mark planes (đ??´đ??ľđ??ś) = đ?›ź, (đ??´đ??ˇđ??ś) = đ?›˝, (đ??ľđ??ˇđ??ś) = đ?›ž, (đ??´đ??ľđ?‘‚) = đ?›ż. Let đ?‘€ be the intersection of the interiors of the 4 trihedral angles. We show that: đ?‘€ = int. [đ??´đ??ľđ??śđ??ˇ], by double inclusion. Ě‚ ∊ int. đ?‘Žđ?‘“đ?‘‘ Ě‚ ∊ int. đ?‘‘đ?‘’đ?‘? Ě‚ ∊ int. đ?‘?đ?‘“đ?‘? Ě‚ â&#x;š đ?‘ƒ ∈ |đ?›źđ??ˇ ∊ |đ?›žđ??ś ∊ đ?›˝đ??ľ and 1. đ?‘ƒ ∈ đ?‘€ â&#x;š đ?‘ƒ ∈ int. đ?‘Žđ?‘?đ?‘? đ?‘ƒ ∈ |đ?›żđ??´ ∊ |đ?›žđ??ś ∊ |đ?›˝đ??ľ â&#x;š đ?‘ƒ ∈ |đ?›źđ??ˇ and đ?‘ƒ ∈ |đ?›˝đ??ľ and đ?‘ƒ ∈ |đ?›žđ??ś and đ?‘ƒ ∈ |đ?›żđ??ś â&#x;š đ?‘ƒ ∈ |đ?›źđ??ˇ ∊ |đ?›žđ??ś ∊ đ?›˝đ??ľ ∊ đ?›żđ??´ â&#x;š đ?‘ƒ ∈ int. [đ??´đ??ľđ??śđ??ˇ]. So đ?‘€ ∈ [đ??´đ??ľđ??śđ??ˇ]. 2. Following the inverse reasoning we show that [đ??´đ??ľđ??śđ??ˇ] ⊂ đ?‘€ from where the equality.

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Solution to Problem 154.

such that such that đ?‘ ∈ |đ??śđ?‘ƒ|, (đ??´đ??ˇđ??ľ) ∊ (đ??ˇđ?‘ƒđ??ś) = đ??ˇđ?‘ƒ. đ?‘™đ?‘’đ?‘šđ?‘šđ?‘Ž

From đ?‘ ∈ |đ??śđ?‘ƒ| and ∈ |đ??ˇđ?‘ | ⇒

int. đ??ˇđ?‘ƒđ??ś â&#x;š đ?‘€ ∈ int. đ??ˇđ?‘ƒđ??ś â&#x;š (∃)đ?‘„ ∈ |đ??ˇđ??ś|.

So we showed that (∃)đ?‘ƒ ∈ |đ??´đ??ľ| and đ?‘„ ∈ |đ??ˇđ??ś| such that đ?‘€ ∈ |đ?‘ƒđ?‘„|.

Solution to Problem 155. Let â„ł be the union of the open segments |đ?‘ƒđ?‘„|. We must prove that: int. [đ??´đ??ľđ??śđ??ˇ] = â„ł through double inclusion. 1. Let đ?‘€ ∈ int. [đ??´đ??ľđ??śđ??ˇ] â&#x;š (∀)đ?‘ƒ ∈ |đ??´đ??ľ and đ?‘„ ∈ |đ??śđ??ˇ such that đ?‘€ ∈ |đ?‘ƒđ?‘„| â&#x;š đ?‘€ ∈ â„ł so int. [đ??´đ??ľđ??śđ??ˇ] ⊂ â„ł. 2. Let đ?‘€ ∈ â„ł â&#x;š (∃)đ?‘ƒ ∈ |đ??´đ??ľ| and đ?‘„ ∈ |đ??śđ??ˇ| such that đ?‘€ ∈ |đ?‘ƒđ?‘„|. Points đ??ˇ, đ??ś and đ?‘ƒ determine plane (đ?‘ƒđ??ˇđ??ś) and (đ?‘ƒđ??ˇđ??ś) ∊ (đ??´đ??śđ??ľ) = đ?‘ƒđ??ś, (đ?‘ƒđ?‘ˆđ??ś) ∊ (đ??´đ??ˇđ??ľ) = đ?‘ƒđ??ˇ. As (∀)đ?‘„ ∈ |đ??śđ??ˇ| such that đ?‘€ ∈ |đ?‘ƒđ?‘„| â&#x;š đ?‘€ ∈ [đ?‘ƒđ??śđ??ˇ] â&#x;š |(∀)đ?‘… ∈ |đ?‘ƒđ??ś| such that đ?‘€ ∈ |đ??ˇđ?‘…|. If đ?‘ƒ ∈ |đ??´đ??ľ| and đ?‘… ∈ |đ?‘ƒđ??ś| â&#x;š đ?‘… ∈ int. đ??´đ??śđ??ľ such that đ?‘€ ∈ |đ??ˇđ?‘…| â&#x;š đ?‘€ int. [đ??´đ??ľđ??śđ??ˇ] â&#x;š â„ł ⊂ int. [đ??´đ??ľđ??śđ??ˇ]. Working with closed segments we obtain that (∀)đ?‘… ∈ [đ??´đ??śđ??ľ] such that đ?‘€ ∈ [đ??ˇđ?‘…], thus obtaining tetrahedron [đ??´đ??ľđ??śđ??ˇ].

Solution to Problem 156. Let đ?‘€ ∈ [đ??´đ??ľđ??śđ??ˇ] â&#x;š (∃) đ?‘ƒ ∈ [đ??´đ??ľđ??ś] such that đ?‘€ ∈ [đ??ˇđ?‘ƒ]. Let đ?‘ ∈ [đ??´đ??ľđ??śđ??ˇ] â&#x;š (∃) đ?‘„ ∈ [đ??´đ??ľđ??ś] such that đ?‘ ∈ [đ??ˇđ?‘„]. The concurrent lines đ??ˇđ?‘€ and đ??ˇđ?‘ determine angle đ??ˇđ?‘€đ?‘ . The surface of triangle đ??ˇđ?‘ƒđ?‘„ is a convex set.

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Let such that đ?‘‚ ∈ [đ??ˇđ?‘…]. But [đ?‘ƒđ?‘„] ⊂ [đ??´đ??ľđ??ś] because đ?‘ƒ ∈ [đ??´đ??ľđ??ś] ∊ đ?‘„ ∈ [đ??´đ??ľđ??ś] and the surface of the triangle is convex. So (∃)đ?‘… ∈ [đ??´đ??ľđ??ś] such that and the tetrahedron is a convex set.

Note: The tetrahedron can be regarded as the intersection of four closed halfspaces which are convex sets.

Solution to Problem 157. Let â„ł be the union of the segments [đ?‘ƒđ?‘„] with đ?‘ƒ ∈ â„ł1 and đ?‘„ ∈ â„ł2. Let đ?‘Ľ, đ?‘Ľâ€™ ∈ â„ł â&#x;š (∀)đ?‘ƒ ∈ â„ł1 and đ?‘„ ∈ â„ł2 such that đ?‘Ľ ∈ [đ?‘ƒđ?‘„]; (∃)đ?‘ƒâ€˛ ∈ â„ł1 and đ?‘„′ ∈ â„ł2 such that đ?‘Ľâ€˛ ∈ [đ?‘ƒâ€˛đ?‘„′]. From đ?‘ƒ, đ?‘ƒâ€˛ ∈ â„ł1 â&#x;š [đ?‘ƒđ?‘ƒâ€˛ ]′ ∈ â„ł1 which is a convex set. From đ?‘„, đ?‘„′ ∈ â„ł2 â&#x;š [đ?‘„đ?‘„′] ∈ â„ł2 which is a convex set. The union of all the segments [đ?‘€đ?‘ ] with đ?‘€ ∈ [đ?‘ƒđ?‘ƒâ€˛] and đ?‘ ∈ [đ?‘„đ?‘„′] is tetrahedron [đ?‘ƒđ?‘ƒâ€˛đ?‘„đ?‘„′] ⊂ â„ł. So from đ?‘Ľ, đ?‘Ľâ€˛ ∈ â„ł â&#x;š |đ?‘Ľđ?‘Ľâ€™| ⊂ â„ł, so set â„ł is convex.

Solution to Problem 158. The interior of the tetrahedron coincides with the union of segments |đ?‘ƒđ?‘„|, đ?‘ƒ ∈ |đ??´đ??ľ| and đ?‘„ ∈ |đ??śđ??ˇ|, that is int. [đ??´đ??ľđ??śđ??ˇ] = {|đ?‘ƒđ?‘„| ∖ đ?‘ƒ ∈ |đ??´đ??ľ|, đ?‘„ ∈ |đ??śđ??ˇ|}. Let’s show that:

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1. Let

2. Let

If we assume đ?‘ ∈ |đ??ˇđ?‘€| â&#x;š |đ??ˇđ?‘€| ∊ (đ??´đ??ľđ??ś) ≠∅ â&#x;š đ?‘€ and đ??ˇ are in different halfspaces in relation to (đ??´đ??ľđ??ś) â&#x;š đ?‘€ ∉ (đ??´đ??ľđ??ś), đ??ˇ, false (it contradicts the hypothesis). So and the second inclusion is proved. As regarding the tetrahedron: [đ??´đ??ľđ??śđ??ˇ] = {[đ?‘ƒđ?‘„] ∖ đ?‘ƒ ∈ [đ??´đ??ľ] and đ?‘„ ∈ [đ??śđ??ˇ]}. 138

255 Compiled and Solved Problems in Geometry and Trigonometry

đ?‘ƒ = đ??´, đ?‘„ ∈ [đ??śđ??ˇ], [đ?‘ƒđ?‘„] describes face [đ??´đ??ˇđ??ś] If đ?‘ƒ = đ??ľ, đ?‘„ ∈ [đ??śđ??ˇ], [đ?‘ƒđ?‘„] describes face [đ??ľđ??ˇđ??ś] đ?‘„ = đ??ś, đ?‘ƒ ∈ [đ??´đ??ľ], [đ?‘ƒđ?‘„] describes face [đ??´đ??ľđ??ś]. Because the triangular surfaces are convex sets and along with their two points đ?‘ƒ, đ?‘„, segment [đ?‘ƒđ?‘„] is included in the respective surface. So, if we add these two situations to the equality from the previous case, we obtain:

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Lines and Planes 159. Let �, � ′ be two parallel lines. If the line � is parallel to a plane �, show that � ′ ||� or �′ ⊂ �. Solution to Problem 159

160. Consider a line �, parallel to the planes � and �, which intersects after the line �. Show that �‖�. Solution to Problem 160

161. Through a given line �, draw a parallel plane with another given line � ′ . Discuss the number of solutions. Solution to Problem 161

162. Determine the union of the lines intersecting a given line đ?‘‘ and parallel to another given line đ?‘‘ ′ (đ?‘‘ âˆŚ đ?‘‘ ′ ). Solution to Problem 162

163. Construct a line that meets two given lines and that is parallel to a third given line. Discuss. Solution to Problem 163

164. If a plane đ?›ź intersects the secant planes after parallel lines, then đ?›ź is parallel to line đ?›˝ ∊ đ?›ž. Solution to Problem 164

165. A variable plane cuts two parallel lines in points đ?‘€ and đ?‘ . Find the geometrical locus of the middle of segment [đ?‘€đ?‘ ]. Solution to Problem 165

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255 Compiled and Solved Problems in Geometry and Trigonometry

166. Two lines are given. Through a given point, draw a parallel plane with both lines. Discuss. Solution to Problem 166

167. Construct a line passing through a given point, which is parallel to a given plane and intersects a given line. Discuss. Solution to Problem 167

168. Show that if triangles đ??´đ??ľđ??ś and đ??´â€˛đ??ľâ€˛đ??śâ€˛, located in different planes, have đ??´đ??ľ âˆĽ đ??´â€˛đ??ľâ€˛, đ??´đ??ś âˆĽ đ??´â€˛đ??śâ€˛ and đ??ľđ??ś âˆĽ đ??ľâ€˛đ??śâ€˛, then lines đ??´đ??´â€˛, đ??ľđ??ľâ€˛, đ??śđ??śâ€˛ are concurrent or parallel. Solution to Problem 168

169. Show that, if two planes are parallel, then a plane intersecting one of them after a line cuts the other one too. Solution to Problem 169

170. Through the parallel lines đ?‘‘ and đ?‘‘′ we draw the planes đ?›ź and đ?›ź ′ distinct from (đ?‘‘, đ?‘‘′ ). Show that đ?›ź âˆĽ đ?›ź ′ or (đ?›ź ∊ đ?›ź ′ ) âˆĽ đ?‘‘. Solution to Problem 170

171. Given a plane đ?›ź, a point đ??´ ∈ đ?›ź and a line đ?‘‘ ⊂ đ?›ź. a. Construct a line đ?‘‘ ′ such that đ?‘‘ ′ ⊂ đ?›ź, đ??´ ∈ đ?‘‘ ′ and đ?‘‘ ′ âˆĽ đ?‘‘. b. Construct a line through đ??´ included in đ?›ź, which forms with đ?‘‘ an angle of a given measure đ?‘Ž. How many solutions are there? Solution to Problem 171

172. Show that relation đ?›ź âˆĽ đ?›˝ defined on the set of planes is an equivalence relation. Define the equivalence classes. Solution to Problem 172

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173. Consider on the set of all lines and planes the relation “đ?‘Ľ âˆĽ đ?‘Śâ€? or đ?‘Ľ = đ?‘Ś, where đ?‘Ľ and đ?‘Ś are lines or planes. Have we defined an equivalence relation? Solution to Problem 173

174. Show that two parallel segments between parallel planes are concurrent. Solution to Problem 174

175. Show that through two lines that are not contained in the same plane, we can draw parallel planes in a unique way. Study also the situation when the two lines are coplanar. Solution to Problem 175

176. Let đ?›ź and đ?›˝ be two parallel planes, đ??´, đ??ľ ∈ đ?›ź, and đ??śđ??ˇ is a parallel line with đ?›ź and đ?›˝. Lines đ??śđ??´, đ??śđ??ľ, đ??ˇđ??ľ, đ??ˇđ??´ cut plane đ?›˝ respectively in đ?‘€, đ?‘ , đ?‘ƒ, đ?‘„. Show that these points are the vertices of a parallelogram. Solution to Problem 176

177. Find the locus of the midpoints of the segments that have their extremities in two parallel planes. Solution to Problem 177

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Solutions Solution to Problem 159.

đ??´âˆˆđ?›ź đ?‘‘ âˆĽ đ?›ź } â&#x;š đ?‘‘′′ âˆĽ đ?›ź Let đ?‘‘′′ âˆĽ đ?‘‘ â&#x;š đ?‘‘′′ âˆĽ đ?›ź or đ?‘‘′ ⊂ đ?›ź. ′ đ?‘‘ âˆĽđ?‘‘ } â&#x;š đ?‘‘′ âˆĽ đ?‘‘ ′′ đ?‘‘′′ âˆĽ đ?‘‘ }

Solution to Problem 160. Let đ??´ ∈ đ?‘Ž ⇒ đ??´ ∈ đ?›ź ∊ đ??´ ∈ đ?›˝. We draw through A, đ?‘‘′ âˆĽ đ?‘‘.

đ?‘‘âˆĽđ?›ź } â&#x;š đ?‘‘′ ⊂ đ?›ź đ?‘‘′ ⊂ đ?›ź ⊂ đ?›˝ đ?‘‘′ âˆĽ đ?‘‘ đ?‘‘′ = đ?‘Ž }â&#x;š }â&#x;š ′ }â&#x;šđ?‘ŽâˆĽđ?‘‘ đ?‘‘âˆĽđ?›˝ đ?›źâˆŠđ?›˝ =đ?‘Ž đ?‘‘ âˆĽđ?‘‘ đ??´ ∈ đ?›˝, ′ } â&#x;š đ?‘‘′ ⊂ đ?›˝ đ?‘‘ âˆĽđ?‘‘

đ??´ ∈ đ?›ź,

Solution to Problem 161. a. If đ?‘‘ âˆŚ đ?‘‘′ there is only one solution and it can be obtained as it follows: Let đ??´ ∈ đ?‘‘. In the plane (đ??´, đ?‘‘′′ ) we draw đ?‘‘′′ âˆĽ đ?‘‘′ . The concurrent lines đ?‘‘ and đ?‘‘′′ determine plane đ?‘Ž. As đ?‘‘′′ âˆĽ đ?‘‘ â&#x;š đ?‘‘ âˆĽ đ?›ź, in the case of the non-coplanar lines. 143

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b. If đ?‘‘ âˆĽ đ?‘‘′ || d'', (∃) infinite solutions. Any plane passing through đ?‘‘ is parallel to đ?‘‘′′ , with the exception of plane (đ?‘‘, đ?‘‘′). c. đ?‘‘ âˆŚ đ?‘‘′, but they are coplanar (∄) solutions.

Solution to Problem 162.

Let đ??´ ∈ đ?‘‘, we draw through đ??´, đ?‘‘1 âˆĽ đ?‘‘′. We write đ?›ź = (đ?‘‘, đ?‘‘1 ). As đ?‘‘1 âˆĽ đ?‘‘′ â&#x;š đ?‘‘′ âˆĽ đ?›ź. Let đ?‘€ ∈ đ?‘‘, arbitrary â&#x;š đ?‘€ ∈ đ?›ź. We draw

đ?›ż âˆĽ đ?›ż ′, đ?‘€ ∈ đ?›ż } â&#x;š đ?›ż ⊂ đ?›ź, so all the parallel lines to đ?‘‘′ intersecting đ?‘‘ are đ?‘‘′ âˆĽ đ?‘Ž

contained in plane đ?›ź. Let đ?›ž ⊂ đ?›ź, đ?›ž âˆĽ đ?‘‘′ â&#x;š đ?›ž ∊ đ?‘‘ = đ??ľ, so (∀) parallel to đ?‘‘′ from đ?›ź intersects đ?‘‘. Thus, the plane đ?›ź represents the required union.

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255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 163.

We draw đ?‘‘ through đ?‘€ such that

đ?‘‘ âˆĽ đ?‘‘′ } â&#x;š đ?‘‘ âˆĽ đ?‘‘3 . According to previous đ?‘‘′ âˆĽ đ?‘‘3

problem: đ?‘‘ ∊ đ?‘‘1 = {đ?‘ }. Therefore,

a. If đ?‘‘3 âˆĽ đ?‘‘1 , the plane đ?›ź is unique, and if đ?‘‘2 ∊ đ?›ź ≠∅, the solution is unique. b. If đ?‘‘1 âˆĽ đ?‘‘3 , (∄)

đ?‘‘ âˆĽ đ?‘‘1 , because it would mean that we can draw through a đ?‘‘ ∊ đ?‘‘1 ≠∅

point two parallel lines đ?‘‘, đ?‘‘1 to the same line đ?‘‘3 . So there is no solution. c. If đ?‘‘1 âˆŚ đ?‘‘3 and đ?‘‘2 ∊ đ?›ź ≠∅, all the parallel lines to đ?‘‘2 cutting đ?‘‘1 are on the plane đ?›ź and none of them can intersect đ?‘‘2 , so the problem has no solution. d. If đ?‘‘2 ⊂ đ?›ź, đ?‘‘1 ∊ đ?‘‘2 ≠∅, let đ?‘‘1 ∊ đ?‘‘2 = {đ?‘‚}, and the required line is parallel to đ?‘‘3 drawn through đ?‘‚ â&#x;š one solution.

e. If đ?‘‘2 ⊂ đ?›ź , đ?‘‘1 âˆĽ đ?‘‘2 . The problem has infinite solutions, (∀) || to đ?‘‘3 which cuts đ?‘‘1 , also cuts đ?‘‘2 .

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Solution to Problem 164.

Solution to Problem 165.

The problem is reduced to the geometrical locus of the midpoints of the segments that have extremities on two parallel lines. đ?‘ƒ is such a point |đ?‘€đ?‘ƒ| = |đ?‘ƒđ?‘ |. Ě‚ = đ??ľđ?‘ƒđ?‘ Ě‚â&#x;š đ?›Ľđ?‘€đ??´đ?‘ƒ = đ?›Ľđ?‘ đ??ľđ?‘ƒ â&#x;š |đ?‘ƒđ??´| ≥ |đ?‘ƒđ??ľ| â&#x;š ||đ??´đ?‘ƒ||? â&#x;š We draw đ??´đ??ľ ⊼ đ?‘‘1 ⇒ đ??´đ??ľ?đ?‘€đ?‘ƒđ??´ the geometrical locus is the parallel to đ?‘‘1 and đ?‘‘2 drawn on the mid-distance between them. It can also be proved vice-versa. 146

255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 166.

Let đ?‘‘1 âˆŚ đ?‘‘2 . In plane (đ?‘‘, đ?‘€) we draw đ?‘‘1′ âˆĽ đ?‘‘1 , đ?‘€ ∈ đ?‘‘1′ . In plane (đ?‘‘2 đ?‘€) we draw đ?‘‘2′ âˆĽ đ?‘‘2 , đ?‘€ ∈ đ?‘‘2′ . We note đ?›ź = đ?‘‘1′ đ?‘‘2′ the plane determined by two concurrent lines.

đ?‘€ ∈ đ?›ź the only solution. Let đ?‘‘1 âˆĽ đ?‘‘2 , đ?‘ ∉ đ?‘‘1 , đ?‘€ ∉ đ?‘‘2 . đ?‘‘1 = đ?‘‘1′ = đ?‘‘2′ In this case đ?‘‘1′ = đ?‘‘2′ = đ?‘‘ and infinite planes pass through đ?‘‘; đ?‘‘2 âˆĽ đ?‘‘ } â&#x;š đ?‘‘1 , đ?‘‘2 are parallel lines with (∀) of the planes passing through đ?‘‘. đ?‘‘1 âˆĽ đ?‘‘ The problem has infinite solutions. But đ?‘€ ∈ đ?‘‘1 or đ?‘€ ∈ đ?‘‘2 , the problem has no solution because the plane can’t pass through a point of a line and be parallel to that line.

Solution to Problem 167. Let đ??´ be the given point, đ?›ź the given plane and đ?‘‘ the given line. a. We assume that đ?‘‘ âˆŚ đ?›ź, đ?‘‘ ∊ đ?›ź = {đ?‘€}. Let plane (đ?‘‘đ??´) which has a common point đ?‘€ with đ?›ź ⇒ (đ?‘‘đ??´) ∊ đ?›ź = đ?‘‘′.

We draw in plane (đ?‘‘đ??´) through point đ??´ a parallel line to đ?‘‘′. 147

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â&#x;š đ?‘Ž is the required line. b. đ?‘‘ âˆĽ đ?›ź, (đ?‘‘đ??´) ∊ đ?›ź ≠∅. Let (dA) ∊ Îą = ι’} ⇒ đ?‘‘′ âˆĽ đ?‘‘ đ?‘‘âˆĽđ?›ź

All the lines passing through đ??´ and intersecting đ?‘‘ are contained in plane (đ?‘‘đ??´). But all these lines also cut đ?‘‘′ âˆĽ đ?‘‘, so they can’t be parallel to đ?›ź. There is no solution. c. đ?‘‘ âˆĽ đ?›ź, (đ?‘‘đ??´) ∊ đ?›ź = ∅.

Let đ?‘€ ∈ đ?‘‘ and line đ??´đ?‘€ ⊂ (đ?‘‘đ??´); (đ?‘‘đ??´) ∊ đ?›ź = ∅ â&#x;š đ??´đ?‘€ ∊ đ?›ź = ∅ â&#x;š đ??´đ?‘€ âˆĽ đ?›ź, (∀)đ?‘€ ∈ đ?‘‘. The problem has infinite solutions.

Solution to Problem 168.

(đ??´đ??ľđ??ś) and (đ??´â€˛đ??ľâ€˛đ??śâ€˛) are distinct planes, thus the six points đ??´, đ??ľ, đ??ś, đ??´â€˛, đ??ľâ€˛, đ??śâ€˛ can’t be coplanar. đ??´đ??ľ âˆĽ đ??´â€˛đ??ľâ€™ â&#x;š đ??´, đ??ľ, đ??´â€˛, đ??ľâ€˛ are coplanar. 148

255 Compiled and Solved Problems in Geometry and Trigonometry

The points are coplanar four by four, that is (đ??´đ??ľđ??ľâ€˛đ??´), (đ??´đ??śđ??śâ€˛đ??´â€˛), (đ??ľđ??śđ??śâ€˛đ??ľâ€˛), and determine four distinct planes. If we assumed that the planes coincide two by two, it would result other 6 coplanar points and this is false. In plane đ??´đ??ľđ??ľâ€˛đ??´â€˛, lines đ??´đ??´â€˛, đ??ľđ??ľâ€˛ can be parallel or concurrent. First we assume that:

is a common point to the 3 distinct planes, but the intersection of 3 distinct planes can be only a point, a line or ∅. It can’t be a line because lines

â&#x;š are distinct if we assumed that two of them coincide, the 6 points would be coplanar, thus there is no common line to all the three planes. There is one possibility left, that is they have a common point đ?‘† and from

We assume đ?‘‘ ∊ đ?›˝ = ∅ â&#x;š đ?‘‘ âˆĽ đ?›˝ â&#x;š đ?‘‘ ∈ plane âˆĽ đ?›˝ drawn through đ??´ â&#x;š đ?‘‘ ⊂ đ?›ź, false. So đ?‘‘ ∊ đ?›˝ = {đ??ľ}.

Solution to Problem 169. Hypothesis: đ?›ź âˆĽ đ?›˝, đ?›ž ∊ đ?›ź = đ?‘‘1 . Conclusion: đ?›ž ∊ đ?›˝ = đ?‘‘2 . We assume that đ?›ž ∊ đ?›˝ = ∅ â&#x;š đ?›ž âˆĽ đ?›˝. Let

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because from a point we can draw only one parallel plane with the given plane. But this result is false, it contradicts the hypothesis đ?›ž ∊ đ?›ź = đ?‘‘1 so đ?›ž ∊ đ?›˝ = đ?‘‘2 .

Solution to Problem 170.

Hypothesis: đ?‘‘ âˆĽ đ?‘‘′; đ?‘‘ ⊂ đ?›ź; đ?‘‘′ ⊂ đ?›źâ€˛; đ?›ź, đ?‘Žâ€˛ ≠(đ?‘‘đ?‘‘ ′ ). Conclusion: đ?›ź âˆĽ đ?›źâ€˛ or đ?‘‘′′ âˆĽ đ?‘‘. As đ?›ź, đ?‘Žâ€˛ ≠(đ?‘‘đ?‘‘′ ) â&#x;š đ?›ź ≠đ?‘Žâ€˛. If đ?‘Ž ∊ đ?‘Žâ€˛ = ∅ â&#x;š đ?‘Ž âˆĽ đ?‘Žâ€˛. If đ?‘Ž ∊ đ?‘Žâ€˛ = ∅ â&#x;š đ?‘Ž ∊ đ?‘Žâ€˛ = đ?‘‘ ′′ . đ?‘‘ âˆĽ đ?‘‘′′ đ?‘‘ âˆĽ đ?‘Žâ€˛ If â&#x;š } â&#x;š đ?‘‘ ′′ ‖đ?‘‘. đ?‘‘⊂đ?‘Ž đ?‘‘′ ⊂ đ?‘Žâ€˛

Solution to Problem 171.

a. If đ??´ ∈ đ?‘‘, then đ?‘‘′ = đ?‘‘. If đ??´ ∉ đ?‘‘, we draw through đ??´, đ?‘‘′ âˆĽ đ?‘‘.

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255 Compiled and Solved Problems in Geometry and Trigonometry ′ Ě‚ b. We draw đ?‘‘1 ⊂ đ?›ź, đ??´ ∈ đ?‘‘, such that đ?‘š(đ?‘‘ 1 đ?‘‘ ) = đ?‘Ž and đ?‘‘ ⊂ đ?›ź, đ??´ ∈ đ?‘‘2 , such that ′ ′ Ě‚ đ?‘š(đ?‘‘ 2 đ?‘‘ ) = đ?‘Ž, a line in each half-plane determined by đ?‘‘ . So (∃) 2 solutions

excepting the situation đ?‘Ž = 0 or đ?‘Ž = 90 when (∃) only one solution.

Solution to Problem 172. đ?›ź âˆĽ đ?›˝ or đ?›ź = đ?›˝ â&#x;ş đ?›ź~đ?›˝ 1. đ?›ź = đ?›ź â&#x;š đ?›ź~đ?›ź, the relation is reflexive; 2. đ?›ź~đ?›˝ â&#x;š đ?›˝~ âˆ?, the relation is symmetric. đ?›ź âˆĽ đ?›˝ or đ?›ź = đ?›˝ â&#x;š đ?›˝ âˆĽ đ?›ź or đ?›˝ = đ?›ź â&#x;š đ?›˝~ âˆ?; 3. đ?›ź~đ?›˝ ∊ đ?›˝~đ?›ž â&#x;š đ?›ź~đ?›ž. If đ?›ź = đ?›˝ ∊ đ?›˝~đ?›ž â&#x;š đ?›ź~đ?›ž. If

đ?›ź ≠đ?›˝ and đ?›ź~đ?›˝ â&#x;š đ?›ź âˆĽ đ?›˝ } â&#x;š đ?›ź âˆĽ đ?›ž â&#x;š đ?›ź~đ?›ž. đ?›˝~đ?›ž â&#x;š đ?›˝ = đ?›ž or đ?›˝ âˆĽ đ?›ž

The equivalence class determined by plane � is constructed of planes �′ with �′~�, that is of � and all the parallel planes with �.

Solution to Problem 173.

No, it is an equivalence relation, because the transitive property is not true. For example, � is a line, � a plane, � a line. From � || � and || � � � || �, lines � and � could be coplanar and concurrent or non-coplanar.

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Solution to Problem 174.

â&#x;š đ??´đ??ľđ??śđ??ˇ parallelogram. So ||đ??´đ??ś|| = ||đ??ľđ??ˇ||.

Solution to Problem 175.

We consider đ??´ ∈ đ?‘‘ and draw through it đ?‘‘1 âˆĽ đ?‘‘′ . We consider đ??ľ ∈ đ?‘‘′ and draw đ?‘‘2 âˆĽ đ?‘‘. Plane (đ?‘‘1 đ?‘‘2 ) âˆĽ (đ?‘‘đ?‘‘1 ), because two concurrent lines from the first plane are parallel with two concurrent lines from the second plane. When đ?‘‘ and đ?‘‘′ are coplanar, the four lines đ?‘‘, đ?‘‘1 , đ?‘‘2 and đ?‘‘′ are coplanar and the two planes coincide with the plane of the lines đ?‘‘ and đ?‘‘′.

Solution to Problem 176. Let planes:

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From (1), (2), (3), (4) â&#x;š đ?‘€đ?‘ đ?‘ƒđ?‘„ parallelogram.

Solution to Problem 177. Let [đ??´đ??ľ] and [đ??śđ??ˇ] be two segments, with đ??´, đ??ś ∈ đ?›ź and đ??ľ, đ??ˇ ∈ đ?›˝ such that |đ??´đ?‘€| = |đ?‘€đ??ľ| and |đ??śđ?‘ | = |đ?‘ đ??ˇ|.

In plane (đ?‘€đ??śđ??ˇ) we draw through đ?‘€, đ??¸đ??š||đ??śđ??ˇ â&#x;š đ??¸đ??ś||đ??ˇđ??š â&#x;š đ??¸đ??šđ??ˇđ??ś parallelogram â&#x;š |đ??¸đ??š| ≥ |đ??śđ??ˇ|. Concurrent lines đ??´đ??ľ and đ??¸đ??š determine a plane which cuts planes đ?›ź after 2 parallel lines â&#x;š đ??¸đ??´||đ??ľđ??š. In this plane, |đ??´đ?‘€| ≥ |đ??ľđ?‘€|. Ě‚ ≥ đ??ľđ?‘€đ??š (angles Ě‚ đ??¸đ?‘€đ??´ opposed at peak) } Ě‚ Ě‚ đ??¸đ??´đ?‘€ ≥ đ??šđ??ľđ?‘€(alternate interior angles)

In parallelogram đ??¸đ??śđ??ˇđ??š,

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So the segment connecting the midpoints of two of the segments with the extremity in đ?›ź and đ?›˝ is parallel to these planes. We also consider [đ??şđ??ť] with đ??ş ∈ đ?›ź, đ??ť ∈ đ?›˝ and |đ??şđ?‘„| ≥ |đ?‘„đ??ť| and we show in the same way that đ?‘‚đ?‘€||đ?›ź and đ?‘‚đ?‘€||đ?›˝. (2) From (1) and (2) â&#x;š đ?‘€, đ?‘ , đ?‘„ are elements of a parallel plane to đ?›ź and đ?›˝, marked by đ?›ž. Vice-versa, let’s show that any point from this plane is the midpoint of a segment, with its extremities in đ?›ź and đ?›˝. Let segment [đ??´đ??ľ] with đ??´ ∈ đ?›ź and đ??ľ ∈ đ?›˝ and |đ??´đ?‘€| = |đ??ľđ?‘€|. Through đ?‘€, we draw the parallel plane with đ?›ź and đ?›˝ and in this plane we consider an arbitrary point đ?‘‚ ∈ đ?›ž. Through đ?‘‚ we draw a line such that đ?‘‘ ∊ đ?›ź = {đ??ź} and đ?‘‘ ∊ đ?›˝ = {đ??ź}. In plane (đ?‘‚đ??´đ??ľ) we draw đ??´â€˛đ??ľâ€˛ || đ??´đ??ľ. Plane (đ??´đ??´â€˛đ??ľâ€˛đ??ľ) cuts the three parallel planes after parallel lines â&#x;š

In plane (đ??´â€˛đ??ľâ€˛đ??ź) â&#x;š |đ??´â€˛đ?‘‚| ≥ |đ?‘‚đ??ľâ€˛| â&#x;š đ??źđ??´â€˛ || đ??ľâ€˛đ??ź and thus Ě‚ â&#x;š đ??źđ??ľâ€˛đ?‘‚

|đ??´â€˛đ?‘‚| ≥ |đ?‘‚đ??ľâ€˛| ̂≥ and đ??źđ??´â€˛đ?‘‚ Ě‚ ≥ đ??źđ?‘‚đ??ľâ€˛ Ě‚ đ??źđ?‘‚đ??´â€˛

� is the midpoint of a segment with extremities in planes � and �. Thus the geometrical locus is plane �, parallel to � and � and passing through the mid-distance between � and �.

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255 Compiled and Solved Problems in Geometry and Trigonometry

Projections 178. Show that if lines đ?‘‘ and đ?‘‘′ are parallel, then prđ?›ź đ?‘‘ âˆĽ prđ?›ź đ?‘‘ ′ or prđ?›ź đ?‘‘ = prđ?›ź đ?‘‘ ′ . What can we say about the projective planes of đ?‘‘ and đ?‘‘ ′ ? Solution to Problem 178

179. Show that the projection of a parallelogram on a plane is a parallelogram or a segment. Solution to Problem 179

180. Knowing that side [đ?‘‚đ??´ of the right angle đ??´đ?‘‚đ??ľ is parallel to a plane đ?›ź, Ě‚ onto the plane đ?›ź is a right angle. show that the projection of đ??´đ?‘‚đ??ľ Solution to Problem 180

181. Let đ??´â€˛đ??ľâ€˛đ??śâ€˛ be the projection of ∆đ??´đ??ľđ??ś onto a plane đ?›ź. Show that the centroid of ∆đ??´đ??ľđ??ś is projected onto the centroid of ∆đ??´â€™đ??ľâ€™đ??śâ€™. Is an analogous result true for the orthocenter? Solution to Problem 181

182. Given the non-coplanar points đ??´, đ??ľ, đ??ś, đ??ˇ, determine a plane on which the points đ??´, đ??ľ, đ??ś, đ??ˇ are projected onto the peaks of parallelogram. Solution to Problem 182

183. Consider all triangles in space that are projected onto a plane � after the same triangle. Find the locus of the centroid. Solution to Problem 183

184. Let đ??´ be a point that is not on line đ?‘‘. Determine a plane đ?›ź such that prđ?›ź đ?‘‘ passes through prđ?›ź đ??´. Solution to Problem 184 155

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185. Determine a plane onto which three given lines to be projected after concurrent lines. Solution to Problem 185

186. Let �, � be planes that cut each other after a line � and let � be a perpendicular line to �. Show that the projections of line � onto �, � are concurrent. Solution to Problem 186

187. Consider lines đ?‘‚đ??´, đ?‘‚đ??ľ, đ?‘‚đ??ś ⊼ two by two. We know that ||đ?‘‚đ??´|| = đ?‘Ž, ||đ?‘‚đ??ľ|| = đ?‘?, ||đ?‘‚đ??ś|| = đ?‘?. Find the measure of the angle of planes (đ??´đ??ľđ??ś) and (đ?‘‚đ??´đ??ľ). Solution to Problem 187

188. A line cuts two perpendicular planes đ?›ź and đ?›˝ at đ??´ and đ??ľ. Let đ??´â€˛ and đ??ľ ′ be the projections of points đ??´ and đ??ľ onto line đ?›ź ∊ đ?›˝. a. Show that ||đ??´đ??ľ||² = ||đ??´đ??´â€˛||² + ||đ??´â€˛đ??ľâ€˛||² + ||đ??ľâ€˛đ??ľ||²; b. If đ?‘Ž, đ?‘?, đ?‘? are the measures of the angles of line đ??´đ??ľ with planes đ?›ź, đ?›˝ and with đ?›ź ∊ đ?›˝, then cos đ?‘?

‖đ??´â€˛ đ??ľâ€˛ ‖ ‖đ??´đ??ľâ€–

and sin2 đ?‘Ž + sin2 đ?‘? = sin2 đ?‘?. Solution to Problem 188

189. Let đ??´đ??ľđ??ś be a triangle located in a plane đ?›ź, đ??´â€˛đ??ľâ€˛đ??śâ€˛ the projection of ∆đ??´â€˛đ??ľâ€˛đ??śâ€˛ onto plane đ?›ź. We mark with đ?‘†, đ?‘† ′ , đ?‘† ′′ the areas of ∆đ??´đ??ľđ??ś, ∆đ??´â€˛ đ??ľ ′ đ??ś ′ , ∆đ??´Ęşđ??ľĘşđ??śĘş, show that đ?‘†â€˛ is proportional mean between đ?‘† and đ?‘† ′′ . Solution to Problem 189

190. A trihedral [đ??´đ??ľđ??śđ??ˇ] has |đ??´đ??ś| ≥ |đ??´đ??ˇ| ≥ |đ??ľđ??ś| ≥ |đ??ľđ??ˇ|. đ?‘€, đ?‘ are the midpoints of edges [đ??´đ??ľ], [đ??śđ??ˇ], show that: a. đ?‘€đ?‘ ⊼ đ??´đ??ľ, đ?‘€đ?‘ ⊼ đ??śđ??ˇ, đ??´đ??ľ ⊼ đ??śđ??ˇ

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b. If đ??´â€˛, đ??ľâ€˛, đ??śâ€˛, đ??ˇâ€˛ are the feet of the perpendicular lines drawn to the peaks đ??´, đ??ľ, đ??ś, đ??ˇ on the opposite faces of the tetrahedron, points đ??ľ, đ??´â€˛, đ?‘ are collinear and so are đ??´, đ??ľâ€˛, đ?‘ ; đ??ˇ, đ??śâ€˛, đ?‘€; đ??ś, đ??ˇâ€˛, đ?‘€. c. đ??´đ??´â€˛, đ??ľđ??ľâ€˛, đ?‘€đ?‘ and đ??śđ??śâ€˛, đ??ˇđ??ˇâ€˛, đ?‘€đ?‘ are groups of three concurrent lines. Solution to Problem 190

191. If rays [đ?‘‚đ??´ and [đ?‘‚đ??ľ with their origin in plane đ?›ź, đ?‘‚đ??´ ⊼ đ?›ź, then the two rays form an acute or an obtuse angle, depending if they are or are not on the same side of plane đ?›ź. Solution to Problem 191

192. Show that the 6 mediator planes of the edges of a tetrahedron have a common point. Through this point pass the perpendicular lines to the faces of the tetrahedron, drawn through the centers of the circles of these faces. Solution to Problem 192

193. Let đ?‘‘ and đ?‘‘′ be two non-coplanar lines. Show that (∃) unique points đ??´ ∈ đ?‘‘, đ??´â€˛ ∈ đ?‘‘′ such that đ??´đ??´â€˛ ⊼ đ?‘‘ and đ??´đ??´â€˛ ⊼ đ?‘‘′. The line đ??´đ??´â€˛ is called the common perpendicular of lines đ?‘‘ and đ?‘‘′. Solution to Problem 193

194. Consider the notations from the previous problem. Let đ?‘€ ∈ đ?‘‘, đ?‘€â€˛ ∈ đ?‘‘′. Show that ‖đ??´đ??´â€˛â€– ≤ ‖đ?‘€đ?‘€â€˛â€–. The equality is possible only if đ?‘€ = đ??´, đ?‘€â€˛ = đ??´â€˛. Solution to Problem 194

195. Let đ??´đ??´â€˛ be the common ⊼ of non-coplanar lines đ?‘‘, đ?‘‘ ′′ and đ?‘€ ∈ đ?‘‘, đ?‘€â€˛ ∈ đ?‘‘′ such that |đ??´đ?‘€| ≥ |đ??´â€˛đ?‘€â€˛|. Find the locus of the midpoint of segment [đ?‘€đ?‘€â€˛]. Solution to Problem 195

196. Consider a tetrahedron đ?‘‰đ??´đ??ľđ??ś with the following properties. đ??´đ??ľđ??ś is an equilateral triangle of side đ?‘Ž, (đ??´đ??ľđ??ś) ⊼ (đ?‘‰đ??ľđ??ś), the planes (đ?‘‰đ??´đ??ś) and (đ?‘‰đ??´đ??ľ)

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form with plane (đ??´đ??ľđ??ś) angles of 60°. Find the distance from point đ?‘‰ to plane (đ??´đ??ľđ??ś). Solution to Problem 196

197. All the edges of a trihedral are of length a. Show that a peak is projected onto the opposite face in its centroid. Find the measure of the dihedral angles determined by two faces. Solution to Problem 197

198. Let đ??ˇđ??¸ be a perpendicular line to the plane of the square đ??´đ??ľđ??śđ??ˇ. Knowing that ‖đ??ľđ??¸â€– = đ?‘™ and that the measure of the angle formed by [đ??ľđ??¸ and (đ??´đ??ľđ??ś) is đ?›˝, determine the length of segment đ??´đ??¸ and the angle of [đ??´đ??¸ with plane (đ??´đ??ľđ??ś). Solution to Problem 198

199. Line đ??śđ??ˇ ⊼ plane of the equilateral ∆đ??´đ??ľđ??ś of side đ?‘Ž, and [đ??´đ??ˇ and [đ??ľđ??ˇ form with plane (đ??´đ??ľđ??ś) angles of measure đ?›˝. Find the angle of planes (đ??´đ??ľđ??ś) and đ??´đ??ľđ??ˇ. Solution to Problem 199

200. Given plane đ?›ź and ∆đ??´đ??ľđ??ś, ∆đ??´â€™đ??ľâ€™đ??śâ€™ that are not on this plane. Determine a ∆đ??ˇđ??¸đ??š, located on đ?›ź such that on one side lines đ??´đ??ˇ, đ??ľđ??¸, đ??śđ??š and on the other side lines đ??´â€˛đ??ˇ, đ??ľâ€˛đ??¸, đ??śâ€˛đ??š are concurrent. Solution to Problem 200

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Solutions Solution to Problem 178. Let đ?‘‘ âˆĽ đ?‘‘′ , đ?›˝ the projective plane of đ?‘‘.

We assume that đ?‘‘ ′ ⊄ đ?›˝, which means that is plane đ?‘‘, đ?‘‘′ ⊼ đ?›ź, â&#x;š the projective plane of đ?‘‘′ is đ?›˝ ′ . We want to show that prđ?‘Ž đ?‘‘ âˆĽ prđ?‘Ž đ?‘‘′. We assume that prđ?‘Ž đ?‘‘ ∊ prđ?‘Ž đ?‘‘′ = {đ?‘ƒ} ⇒ (∃) đ?‘€ ∈ đ?‘‘ such that prđ?‘Ž đ?‘€ = đ?‘ƒ and (∃)đ?‘€â€˛ ∈ đ?‘‘′ such that prđ?‘Ž đ?‘€â€˛ = đ?‘ƒ. đ?‘ƒđ?‘€ ⊼ đ?›ź } â&#x;š in the point đ?‘ƒ on plane đ?›ź we can draw two distinct perpendicular đ?‘ƒđ?‘€â€˛ ⊼ đ?›ź lines. False. â&#x;š

If đ?›˝ is the projective plane of đ?‘‘ and đ?›˝ of đ?‘‘′ , then đ?›˝ âˆĽ đ?›˝ ′ , because if they had a common point their projections should be elements of prđ?‘Ž đ?‘‘ and prđ?‘Ž đ?‘‘′ , and thus they wouldn’t be anymore parallel lines. If đ?‘‘′ ⊂ đ?›˝ or đ?‘‘ ⊂ đ?›˝ ′ , that is (đ?‘‘, đ?‘‘′ ) ⊼ đ?›ź â&#x;š đ?‘‘ and đ?‘‘′ have the same projective plane â&#x;š prđ?‘Ž đ?‘‘ = prđ?‘Ž đ?‘‘′ .

Solution to Problem 179.

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We assume that đ??´đ??ľđ??śđ??ˇ ⊼ đ?›ź. Let đ??´â€˛, đ??ľâ€˛, đ??śâ€˛, đ??ˇâ€˛ be the projections of points đ??´, đ??ľ, đ??ś, đ??ˇ. đ?‘?đ?‘&#x;1

đ??´đ??ľ âˆĽ đ??ˇđ??ś ⇒ đ??´â€˛ đ??ľâ€˛ âˆĽ đ??ˇ ′ đ??ś ′ } â&#x;š đ??´â€˛, đ??ľâ€˛, đ??śâ€˛, đ??ˇâ€˛ parallelogram. đ??´đ??ˇ âˆĽ đ??ľđ??ś â&#x;š đ??´â€˛ đ??ˇ ′ âˆĽ đ??ľâ€˛ đ??ˇ ′ If (đ??´đ??ľđ??śđ??ˇ) ⊼ đ?›ź ⇒ the projection đ??´â€˛, đ??ľâ€˛, đ??śâ€˛, đ??ˇ ′ ∈ the line (đ??´đ??ľđ??śđ??ˇ) ∊ đ?›ź â&#x;š the projection of the parallelogram is a segment.

Solution to Problem 180.

If đ?‘‚đ??´||đ?›ź â&#x;š projđ?›ź đ?‘‚đ??´||đ?‘‚đ??´ â&#x;š đ?‘‚′đ??´â€˛||đ?‘‚đ??´ because (∀) a plane which passes through đ?‘‚đ??´ cuts the plane đ?›ź after a parallel to đ?‘‚đ??´.

Ě‚ is a right angle. â&#x;š đ?‘‚′đ??´â€˛ ⊼ đ?‘‚′đ??ľâ€˛ â&#x;š đ??´â€˛đ?‘‚′đ??ľâ€˛

Solution to Problem 181.

In the trapezoid đ??ľđ??śđ??śâ€˛đ??ľâ€˛ (đ??ľđ??ľâ€˛||đ??śđ??śâ€˛),

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â&#x;š đ??´â€˛đ?‘€â€˛ is a median.

đ?‘€đ?‘€â€˛ âˆĽ đ??´đ??´â€˛ â&#x;š đ?‘€đ?‘€â€˛đ??´â€˛đ??´ trapezoid ‖đ??´â€˛đ??şâ€– ‖đ??´đ??şâ€– ‖đ??´đ??şâ€– } = =2 ‖đ??şâ€˛đ?‘€â€˛â€– ‖đ??şđ?‘€â€– = 2, đ??şđ??şâ€˛ âˆĽ đ??´đ??´â€˛ ‖đ??şđ?‘€â€– â&#x;š đ??şâ€˛ is on median A'M' at 2/3 from the peak and 1/3 from the base.

Generally no, because the right angle đ??´đ?‘€đ??ś should be projected after a right angle. The same thing is true for another height. This is achieved if the sides of the ∆ are parallel to the plane.

Solution to Problem 182.

Let đ??´, đ??ľ, đ??ś, đ??ˇ be the 4 non-coplanar points and đ?‘€, đ?‘ midpoints of segments |đ??´đ??ľ| and |đ??śđ??ˇ|. đ?‘€ and đ?‘ determine a line and let a plane đ?›ź ⊼ đ?‘€đ?‘ , đ?‘€ and đ?‘ are projected in the same point đ?‘‚ onto đ?›ź.

â&#x;š đ??´â€˛đ??ľâ€˛đ??śâ€˛đ??ˇâ€˛ a parallelogram.

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Solution to Problem 183.

Let đ??´â€˛đ??ľâ€˛đ??śâ€˛, đ??´â€˛â€˛đ??ľâ€˛â€˛đ??śâ€˛â€˛ two triangles of this type, with the following property:

â&#x;š đ??ş ′ đ??ş = đ??şđ??ş ′′ â&#x;š đ??ş ′′ , đ??ş ′ , đ??ş are collinear. Due to the fact that by projection the ratio is maintained, we show that đ??şâ€˛â€˛ is the centroid of đ??´, đ??ľ, đ??ś.

Solution to Problem 184. Let đ?‘€ ∈ đ?‘‘ and đ??´ ∉ đ?‘‘. The two points determine a line and let đ?›ź be a perpendicular plane to this line, đ??´đ?‘€ ⊼ đ?›ź â&#x;š đ??´ and đ?‘€ are projected onto đ?›ź in the same point đ??´â€˛ through which also passes projÎą đ?‘‘ = projÎą đ??´ ∈ projÎą đ?‘‘.

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255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 185. We determine a line which meets the three lines in the following way. Let

Let now a plane

Solution to Problem 186.

Let đ?›ź ∊ đ?›˝ = đ?‘Ž and đ?‘€ ∈ đ?‘‘. We project this point onto đ?›ź and đ?›˝:

â&#x;š đ?‘Ž ⊼ onto the projective plane of đ?‘‘ onto đ?›˝.

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Let

â&#x;š đ?‘‚đ?‘€â€˛ ∊ đ?‘‚đ?‘€â€˛â€˛ = {đ?‘‚}, so the two projections are concurrent.

Solution to Problem 187.

Ě‚ = đ?›ź. the angle of planes (đ??´đ??ľđ??ś) and (đ?‘‚đ??´đ??ľ) is đ?‘‚đ?‘€đ??ś

Solution to Problem 188. Let đ?›ź ∊ đ?›˝ = a and đ??´đ??´â€˛ ⊼ đ?‘Ž, đ??ľđ??ľâ€˛ ⊼ đ?‘Ž.

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255 Compiled and Solved Problems in Geometry and Trigonometry

As

Ě‚ =đ?‘Ž â&#x;š âˆ˘ of line đ??´đ??ľ with đ?›ź, đ??ľđ??´đ??ľâ€˛ â&#x;š âˆ˘ of line đ??´đ??ľ with đ?›˝, đ??´đ??ľđ??´â€˛ = đ?‘? In the plane đ?›˝ we draw through đ??ľ a parallel line to đ?‘Ž and through đ??´â€˛ a parallel line to đ??ľđ??ľâ€˛ . Their intersection is đ??ś, and ||đ??´â€˛ đ??ľâ€˛|| = ||đ??ľđ??ś||, ||đ??ľđ??ľâ€˛|| = ||đ??´â€˛ đ??ś||. The angle of line Ě‚ = đ?‘?. đ??´đ??ľ with đ?›ź is đ??´đ??ľđ??ś As đ??´đ??´â€™ ⊼ đ?›˝ â&#x;š đ??´đ??´â€˛ ⊼ đ??´â€˛ đ??ś â&#x;š ‖đ??´đ??śâ€–2 = ‖đ??´đ??´â€˛ ‖2 + ‖đ??´â€˛đ??śâ€–2 = ‖đ??´đ??´â€˛â€–2 + ‖đ??ľâ€˛đ??ľâ€–2 (1) đ??ľâ€˛ đ??ľđ??śđ??´ rectangle

â&#x;š ∆đ??´đ??śđ??ľ is right in đ??ś. We divide the relation (1) with ||đ??´đ??ľ||²:

Solution to Problem 189.

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Solution to Problem 190.

|đ??´đ??ś| ≥ |đ??ľđ??ś| â&#x;š ∆đ??´đ??śđ??ľ isosceles } â&#x;š đ??śđ?‘€ ⊼ đ??´đ??ľ (1) đ??śđ?‘€ median

a.

|đ??´đ??ˇ| ≥ |đ??ľđ??ś| â&#x;š ∆đ??´đ??ľđ??ˇ isosceles } â&#x;š đ??ˇđ?‘€ ⊼ đ??´đ??ľ (2) đ??ˇđ?‘€ median From (1) and (2) â&#x;š đ??´đ??ľ ⊼ (đ??ˇđ?‘€đ??ś) =

đ??´đ??ľ ⊼ đ?‘€đ?‘ } đ??´đ??ľ ⊼ đ??ˇđ??ś

|đ??ľđ??ś| ≥ |đ??ľđ??ˇ| } â&#x;š đ??ľđ?‘ ⊼ đ??ˇđ??ś đ??ľđ?‘ median } â&#x;š đ??ˇđ??ś ⊼ (đ??´đ??ľđ?‘ ) â&#x;š đ??ˇđ??ś ⊼ đ?‘€đ?‘ |đ??´đ??ˇ| ≥ |đ??´đ??ś| } â&#x;š đ??´đ?‘ ⊼ đ??ˇđ??ś đ??´đ?‘ median b.

â&#x;š đ??´â€˛ ∈ đ??ľđ?‘ â&#x;š đ??ľ, đ??´â€˛ , đ?‘ are collinear. In the same way: From

(đ??´đ??ˇđ??ś) ⊼ (đ??´đ??ľđ?‘ ) â&#x;š đ??´, đ??ľâ€˛, đ?‘ collinear (đ??´đ??ľđ??ś) ⊼ (đ??ˇđ?‘€đ??ś) â&#x;š đ?‘€, đ??ˇâ€˛, đ??ś collinear (đ??´đ??ľđ??ˇ) ⊼ (đ??ˇđ?‘€đ??ś) â&#x;š đ??ˇ, đ??śâ€˛, đ?‘€ collinear

C. At point a. we’ve shown that đ?‘€đ?‘ ⊼ đ??´đ??ľ

đ??´đ??´â€˛, đ??ľđ??ľâ€˛ and đ?‘€đ?‘ are heights in ∆đ??´đ??ľđ?‘ , so they are concurrent lines. In the same way, đ??ˇđ??ˇâ€˛, đ??śđ??śâ€˛, đ?‘€đ?‘ will be heights in ∆đ??ˇđ?‘€đ??ś.

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255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 191.

We assume that [đ?‘‚đ??´, [đ?‘‚đ??ľ are on the same side of plane đ?›ź. We draw

(∃) plane (đ??´đ??ˇ, đ??ľđ??ľâ€˛) = đ?›˝ â&#x;š |đ?‘‚đ??´, |đ?‘‚đ??ľ are in the same half-plane.

Ě‚ ) = 900 = đ?‘š(đ??ľđ?‘‚đ??ľ Ě‚ ′) < 900 â&#x;š đ??´đ?‘‚đ??ľ Ě‚ acute. In plane đ?›˝ we have đ?‘š(đ??´đ?‘‚đ??ľ We assume that [đ?‘‚đ??´ and [đ?‘‚đ??ľ are in different half-planes in relation to đ?›ź â&#x;š đ??´ and đ??ľ are Ě‚ â&#x;š đ?‘š(đ??´đ?‘‚đ??ľ Ě‚) = in different half-planes in relation to đ?‘‚đ??ľâ€˛ in plane đ?›˝ â&#x;š |đ?‘‚đ??ľâ€™ ⊂ int. đ??ľđ?‘‚đ??´ Ě‚ ) > 900 â&#x;š đ??´đ?‘‚đ??ľ obtuse. 900 + đ?‘š(đ??ľđ?‘‚đ??ľâ€™

Solution to Problem 192.

We know the locus of the points in space equally distant from the peaks of ∆đ??ľđ??śđ??ˇ is the perpendicular line đ?‘‘ to the pl. ∆ in the center of the circumscribed circle of this ∆, marked with đ?‘‚. We draw the mediator plane of side |đ??´đ??ś|, which intersects this ⊼ đ?‘‘ at point đ?‘‚. Then, point đ?‘‚ is equally distant from all the peaks of the tetrahedron ||đ?‘‚đ??´|| = ||đ?‘‚đ??ľ|| = ||đ?‘‚đ??ś|| = ||đ?‘‚đ??ˇ||. We connect đ?‘‚ with midpoint đ??¸ of side |đ??´đ??ľ|. From |đ?‘‚đ??´| ≥ |đ?‘‚đ??ľ| â&#x;š ∆đ?‘‚đ??´đ??ľ isosceles â&#x;š đ?‘‚đ??ś ⊼ đ??´đ??ľ (1). 167

Florentin Smarandache

We project đ?‘‚ onto plane (đ??´đ??ľđ??ˇ) in point đ?‘‚2 . |đ?‘‚đ??´| ≥ |đ?‘‚đ??ľ| ≥ |đ?‘‚đ??ˇ| As } â&#x;š ∆đ?‘‚đ??´đ?‘‚2 = ∆đ?‘‚đ??ľđ?‘‚2 = ∆đ?‘‚đ??ˇđ?‘‚2 |đ?‘‚)đ?‘‚2 common side â&#x;š |đ?‘‚2 đ??´| ≥ |đ??ľđ?‘‚2 | ≥ |đ??ˇđ?‘‚2 | â&#x;š â&#x;š đ?‘‚2 is the center of the circumscribed circle of ∆đ??´đ??ľđ??ˇ. We show in the same way that đ?‘‚ is also projected on the other faces onto the centers of the circumscribed circles, thus through đ?‘‚ pass all the perpendicular lines to the faces of the tetrahedron. These lines are drawn through the centers of the circumscribed circles. So b. is proved. From |đ?‘‚2 đ??´| ≥ |đ?‘‚đ??ľ2 | â&#x;š ∆đ?‘‚2 đ??´đ??ľ isosceles đ?‘‚2 đ??¸ ⊼ đ??´đ??ľ (2) đ??´đ??ľ ⊼ (đ??¸đ?‘‚2 đ?‘‚) From (1) and (2) â&#x;š }â&#x;š |đ??´đ??¸| ≥ |đ??¸đ??ľ| â&#x;š (đ??¸đ?‘‚2 đ?‘‚) is a mediator plane of side |đ??´đ??ľ| and passes through đ?‘‚ and the intersection of the 3 mediator planes of sides |đ??ľđ??ś|, |đ??śđ??ˇ|, |đ??ľđ??ˇ| belongs to line đ?‘‘, thus O is the common point for the 6 mediator planes of the edges of a tetrahedron.

Solution to Problem 193.

Let đ?‘€ ∈ đ?‘‘ and đ?›ż||đ?‘‘′, đ?‘€ ∈ đ?›ż. Let = (đ?‘‘, đ?›ż) â&#x;š đ?‘‘′||đ?›ź. Let

= {đ??´} otherwise đ?‘‘ and đ?‘‘′ would be parallel, thus coplanar. Let đ?›˝ be the projective plane of line

In plane đ?›˝ we construct a perpendicular to đ?‘‘′′ in point đ??´ and

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255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 194.

We draw

We can obtain the equality only when đ?‘€ = đ??´ and đ?‘€â€˛ = đ??´â€˛.

Solution to Problem 195. Let đ?‘€ ∈ đ?‘‘, đ?‘€â€˛ ∈ đ?‘‘′ such that |đ??´đ?‘€| ≥ |đ??´â€˛đ?‘€â€˛|. Let đ?‘‘′′ = prđ?›ź đ?‘‘′ and đ?‘€â€˛đ?‘€â€˛â€˛ ⊼ đ?‘‘′′ â&#x;š đ?‘€â€˛đ?‘€â€˛â€˛ ⊼ đ?›ź â&#x;š đ?‘€â€˛đ?‘€â€˛â€˛ ⊼ đ?‘€â€˛â€˛đ?‘€.

â&#x;š ∆đ??´đ?‘€đ?‘€â€˛ isosceles. Let đ?‘ƒ be the midpoint of |đ?‘€đ?‘€â€˛| and đ?‘ƒâ€˛ = prđ?›ź đ?‘ƒ â&#x;š đ?‘ƒđ?‘ƒâ€˛ âˆĽ đ?‘€â€˛đ?‘€â€˛â€˛ â&#x;š đ?‘ƒâ€˛ is the midpoint Ě‚ . (đ?‘ƒđ?‘ƒâ€˛) is midline in of đ?‘€đ?‘€â€˛â€˛, ∆đ??´đ?‘€đ?‘€â€˛â€˛ isosceles â&#x;š [đ??´đ?‘ƒâ€˛ the bisector of đ?‘€â€˛đ??´đ?‘€ 1

1

∆đ?‘€â€˛ đ?‘€Ęşđ?‘€ â&#x;š ‖đ?‘ƒđ?‘ƒâ€˛ ‖ = ‖đ?‘€â€˛ đ?‘€â€˛â€˛ = ‖đ??´â€˛đ??´â€– = constant. 2 2 Thus, the point is at a constant distance from line đ??´đ?‘ƒâ€˛, thus on a parallel line to this line, located in the ⊼ plane đ?›ź, which passes through đ??´đ?‘ƒâ€˛. When đ?‘€ = đ??´ and đ?‘€â€˛ = đ??´â€˛ â&#x;š ||đ??´đ?‘€|| = ||đ?‘ ′đ??´â€˛ = 0 â&#x;š đ?‘ƒ = đ?‘…, where đ?‘… is the midpoint of segment |đ??´đ??´â€˛|. So the locus passes through đ?‘… and because

â&#x;š đ?‘…đ?‘ƒ is contained in the mediator plane of segment |đ??´đ??´â€˛|. So đ?‘…đ?‘ƒ is the intersection of the mediator plane of segment |đ??´đ??´â€˛| with the ⊼ plane to đ?›ź, passing through one of the bisectors of the angles determined by đ?‘‘ and đ?‘‘′, we obtain one more line contained by the mediator plane of [đ??´đ??´â€˛], the parallel line with the other bisector of the angles determined by đ?‘‘ and đ?‘‘′′. So the locus will be formed by two perpendicular lines. 169

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Vice-versa, let đ?‘„ ∈ đ?‘…đ?‘ƒ a (∀) point on this line and đ?‘„’ = prđ?›ź đ?‘„ â&#x;š đ?‘„′ ∈ |đ??´đ?‘ƒâ€˛ bisector. We draw đ?‘ đ?‘ ′′ ⊼ đ??´đ?‘„′ and because đ??´đ?‘„′ is both bisector and height â&#x;š ∆đ??´đ?‘ đ?‘ ′′ isosceles â&#x;š |đ??´đ?‘„′| median â&#x;š |đ?‘ đ?‘„′| ≥ |đ?‘„′đ?‘ ′′|. We draw

coplanar

As

â&#x;š |đ?‘„′đ?‘„| midline in ∆đ?‘ đ?‘ ′ đ?‘ ′′ â&#x;š đ?‘„, đ?‘ ′ , đ?‘ collinear and |đ?‘„đ?‘ ′| ≥ |đ?‘„đ?‘ |.

Solution to Problem 196.

where Îą = (ABC).

VD common side

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255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 197.

Let

||VO|| common

â&#x;š đ?‘‚ is the center of the circumscribed circle and as ∆đ??´đ??ľđ??ś is equilateral â&#x;š đ?‘‚ is the centroid â&#x;š

Solution to Problem 198.

In ∆đ??´đ??¸đ??ľ, right in đ??´:

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Solution to Problem 199. đ??śđ??¸ ⊼ đ??ľđ??´ Ě‚ ). } â&#x;š âˆ˘pl. (đ??´đ??ľđ??ś) and đ??´đ??ľđ??ˇ are đ?‘š(đ??ˇđ??¸đ??ś đ??ˇđ??¸ ⊼ đ??´đ??ľ đ?‘Žâˆš3 đ??´đ??ľđ??ś equilateral â&#x;š ‖đ??śđ??¸â€– = . 2

Solution to Problem 200. We consider the problem solved and we take on plane đ?›ź, ∆đ??ˇđ??¸đ??š, then points đ?‘‚ and đ?‘‚′ which are not located on đ?›ź. We also construct lines |đ??ˇđ?‘‚, |đ??šđ?‘‚, |đ??¸đ?‘‚ respectively |đ??ˇđ?‘‚′, |đ??šđ?‘‚′, |đ??¸đ?‘‚′. On these rays we take ∆đ??´đ??ľđ??ś and ∆đ??´â€˛đ??ľâ€˛đ??śâ€˛. Obviously, the way we have constructed the lines đ??´đ??ˇ, đ??ľđ??¸, đ??śđ??š shows that they intersect at đ?‘‚. We extend lines đ??ľđ??´, đ??ľđ??ś, đ??śđ??´ until they intersect plane

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255 Compiled and Solved Problems in Geometry and Trigonometry

đ?›ź at points đ??ľ, đ??ś respectively đ??´. Then, we extend lines đ??śâ€˛đ??´â€˛, đ??śâ€˛đ??ľâ€˛, đ??´â€˛đ??ľâ€˛ until they intersect plane đ?›ź at points đ??´2 , đ??ś2 respectively đ??ľ2 . Obviously, points đ??´1 , đ??ľ1 , đ??ś1 are collinear (because ∈ đ?›ź ∊ (đ??´đ??ľđ??ś)) and đ??´2 , đ??ľ2 , đ??ś2 are as well collinear (because ∈ đ?›ź ∊ (đ??´â€˛đ??ľâ€˛đ??śâ€˛)). On the other side, points đ??ˇ, đ??š, đ??´1 , đ??´2 are collinear because:

thus collinear (1)

â&#x;š đ??ˇ, đ??š, đ??´2 collinear (2) From (1) and (2) â&#x;š đ??ˇ, đ??š, đ??´1 , đ??´2 collinear. Similarly đ??ś, đ??¸, đ??š, đ??ś2 collinear and đ??ľ1 , đ??¸, đ??ˇ, đ??ˇ2 collinear. Consequently, đ??ˇđ??¸đ??š is at the intersection of lines đ??´1 đ??´2 , đ??ś1 đ??ś2 , đ??ľ1 đ??ľ2 on plane đ?›ź, thus uniquely determined.

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Review Problems 201. Find the position of the third peak of the equilateral triangle, the affixes of two peaks being �1 = 1, �2 = 2 + �. Solution to Problem 201

202. Let đ?‘§1 , đ?‘§2 , đ?‘§3 be three complex numbers, not equal to 0, + two by 2, and of equal moduli. Prove that if đ?‘§1 + đ?‘§2 đ?‘§3 , đ?‘§2 + đ?‘§3 đ?‘§1 , đ?‘§2 + đ?‘§ 1 đ?‘§3 ∈ đ?‘… ⇒ đ?‘§1 đ?‘§2 đ?‘§3 = 1. Solution to Problem 202

203. We mark by đ??ş the set of đ?‘› roots of the unit, đ??ş = {đ?œ€0 , đ?œ€1 , ‌ , đ?œ€đ?‘›âˆ’1 }. Prove that: a. đ?œ€đ?‘– ∙ đ?œ€đ?‘— ∈ đ??ş, (∀) đ?‘–, đ?‘— ∈ {0, 1, ‌ , đ?‘› − 1}; b. đ?œ€đ?‘–−1 ∈ đ??ş, (∀ ) đ?‘– ∈ {0, 1, ‌ , đ?‘› − 1}. Solution to Problem 203

204. Let the equation đ?‘Žđ?‘§Â˛ + đ?‘?đ?‘§Â˛ + đ?‘? = 0, đ?‘Ž, đ?‘?, đ?‘? ∈ đ??ś and argđ?‘Ž + argđ?‘? = 2argđ?‘?, and |đ?‘Ž| + |đ?‘?| = |đ?‘?|. Show that the given equation has at list one root of unity. Solution to Problem 204

205. Let đ?‘§1 , đ?‘§2 , đ?‘§3 be three complex numbers, not equal to 0, such that |đ?‘§1 | = |đ?‘§2 | = |đ?‘§3 |. a. Prove that (∃) complex numbers đ?›ź and đ?›˝ such that đ?‘§2 = đ?›źđ?‘§1 , đ?‘§3 = đ?›˝đ?‘§2 and |đ?›ź| = |đ?›˝| = 1; b. Solve the equation đ?›źÂ˛ + đ?›˝Â˛â€“ đ?›ź ∙ đ?›˝â€“ đ?›źâ€“ đ?›˝ + 1 = 0 in relation to one of the unknowns. c. Possibly using the results from đ?‘Ž. and đ?‘?., prove that if đ?‘§12 + đ?‘§22 + đ?‘§32 = đ?‘§1 đ?‘§2 + đ?‘§2 đ?‘§3 + đ?‘§1 đ?‘§3 , then we have đ?‘§1 = đ?‘§2 = đ?‘§3 or the numbers đ?‘§1 , đ?‘§2 , đ?‘§3 are affixes of the peaks of an equilateral ∆. Solution to Problem 205

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255 Compiled and Solved Problems in Geometry and Trigonometry

206. Draw a plane through two given lines, such that their line of intersection to be contained in a given plane. Solution to Problem 206

207. Let đ?‘Ž, đ?‘?, đ?‘? be three lines with a common point and đ?‘ƒ a point not located on any of them. Show that planes (đ?‘ƒđ?‘Ž), (đ?‘ƒđ?‘?), (đ?‘ƒđ?‘?) contain a common line. Solution to Problem 207

208. Let đ??´, đ??ľ, đ??ś, đ??ˇ be points and đ?›ź a plane separating points đ??´ and đ??ľ, đ??´ and đ??ś, đ??ś and đ??ˇ. Show that đ?›ź ∊ |đ??ľđ??ˇ| ≠∅ and đ?›ź ∊ |đ??´đ??ˇ| = ∅. Solution to Problem 208

209. On edges đ?‘Ž, đ?‘?, đ?‘? of a trihedral angle with its peak đ?‘‚, take points đ??´, đ??ľ, đ??ś; let then đ??ˇ ∈ |đ??ľđ??ś| and đ??¸ ∈ |đ??´đ??ˇ|. Show that |đ?‘‚đ??¸ ⊂ đ?‘–đ?‘›đ?‘Ą. ∠đ?‘Žđ?‘?đ?‘?. Solution to Problem 209

210. Show that the following sets are convex: the interior of a trihedral angle, a tetrahedron without an edge (without a face). Solution to Problem 210

211. Let đ??´, đ??ľ, đ??ś, đ??ˇ be four non-coplanar points and đ??¸, đ??š, đ??ş, đ??ť the midpoints of segments [đ??´đ??ľ], [đ??ľđ??ś], [đ??śđ??ˇ], [đ??ˇđ??´]. Show that đ??¸đ??š || (đ??´đ??śđ??ˇ) and points đ??¸, đ??š, đ??ş, đ??ť are coplanar. Solution to Problem 211

212. On lines đ?‘‘, đ?‘‘′ consider distinct points đ??´, đ??ľ, đ??ś; đ??´â€˛, đ??ľâ€˛, đ??śâ€˛ respectively. Show that we can draw through lines đ??´đ??´â€˛, đ??ľđ??ľâ€˛, đ??śđ??śâ€˛ three parallel planes if and only if

‖đ??´đ??ľâ€– ‖đ??´â€˛ đ??ľâ€˛ ‖

‖đ??ľđ??śâ€–

= ‖đ??ľâ€˛ đ??ś ′ ‖ . Solution to Problem 212

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213. Let đ?‘€, đ?‘€â€˛ be each mobile points on the non-coplanar lines đ?‘‘, đ?‘‘′. Find the locus of points đ?‘ƒ that divide segment |đ?‘€đ?‘€â€˛| in a given ratio. Solution to Problem 213

214. Construct a line that meets three given lines, respectively in đ?‘€, đ?‘ , đ?‘ƒ and for which

‖đ?‘€đ?‘ ‖ ‖đ?‘ đ?‘ƒâ€–

to be given ratio. Solution to Problem 214

215. Find the locus of the peak đ?‘ƒ of the triangle đ?‘€, đ?‘ , đ?‘ƒ if its sides remain parallel to three fixed lines, the peak đ?‘€ describes a given line đ?‘‘, and the peak đ?‘ ∈ a given plane đ?›ź. Solution to Problem 215

216. On the edges [đ?‘‚đ??´, [đ?‘‚đ??ľ, [đ?‘‚đ??ś of a trihedral angle we consider points đ?‘€, đ?‘ , đ?‘ƒ such that ‖đ?‘‚đ?‘€â€– = đ?œ†â€–đ?‘‚đ??´â€–, ‖đ?‘‚đ?‘ ‖ = đ?œ†â€–đ?‘‚đ??ľâ€–, ‖đ?‘‚đ?‘ƒâ€– = đ?œ†â€–đ?‘‚đ??śâ€–, where đ?œ† is a positive variable number. Show the locus of the centroid of triangle đ?‘€đ?‘ đ?‘ƒ. Solution to Problem 216

217. đ??´đ??ľđ??śđ??ˇ and đ??´1 đ??ľ1 đ??ś1 đ??ˇ1 are two parallelograms in space. We take the points đ??´2 , đ??ľ2 , đ??ś2 , đ??ˇ2 which divide segments [đ??´đ??´1 ], [đ??ľđ??ľ1 ], [đ??śđ??ś1 ], [đ??ˇđ??ˇ1 ] in the same ratio. Show that đ??´2 đ??ľ2 đ??ś2 đ??ˇ2 is a parallelogram. Solution to Problem 217

218. The lines đ?‘‘, đ?‘‘′ are given, which cut a given plane đ?›ź in đ??´ and đ??´â€˛ . Construct the points đ?‘€, đ?‘€â€˛ on đ?‘‘, đ?‘‘′ such that đ?‘€đ?‘€â€˛ âˆĽ đ?›ź and segment [đ?‘€đ?‘€â€˛] to have a given length đ?‘™. Discuss. Solution to Problem 218

219. Construct a line which passes through a given point đ??´ and that is perpendicular to two given lines đ?‘‘ and đ?‘‘′. Solution to Problem 219

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255 Compiled and Solved Problems in Geometry and Trigonometry

220. Show that there exist three lines with a common point, perpendicular two by two. Solution to Problem 220

221. Let đ?‘Ž đ?‘?, đ?‘?, đ?‘‘ four lines with a common point, đ?‘‘ is perpendicular to đ?‘Ž đ?‘?, đ?‘?. Show that lines đ?‘Ž, đ?‘?, đ?‘? are coplanar. Solution to Problem 221

222. Show that there do not exist four lines with a common point that are perpendicular two by two. Solution to Problem 222

223. Let đ?‘‘ ⊼ đ?›ź and đ?‘‘ ′ âˆĽ đ?‘‘. Show that đ?‘‘′ ⊼ đ?›ź. Solution to Problem 223

224. Show that two distinct perpendicular lines on a plane are parallel. Solution to Problem 224

225. Let đ?‘‘ ⊼ đ?›ź and đ?‘‘ ′ [âˆĽ đ?›ź. Show that đ?‘‘ ′ ⊼ đ?‘‘. Solution to Problem 225

226. Show that two perpendicular planes on the same line are parallel with each other. Solution to Problem 226

227. Show that the locus of the points equally distant from two distinct points đ??´ and đ??ľ is a perpendicular plane to đ??´đ??ľ, passing through midpoint đ?‘‚ of the segment [đ??´đ??ľ] (called mediator plane of [đ??´đ??ľ]). Solution to Problem 227

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228. Find the locus of the points in space equally distant from the peaks of a triangle đ??´đ??ľđ??ś. Solution to Problem 228

229. The plane đ?›ź and the points đ??´ ∈ đ?›ź, đ??ľ ∉ đ?›ź are given. A variable line đ?‘‘ passes through đ??´ and it is contained in plane đ?›ź. Find the locus of the ⊼ feet from đ??ľ to đ?‘‘. Solution to Problem 229

230. A line đ?›ź, and a point đ??´ ∉ đ?›ź are given. Find the locus of the feet of the perpendicular lines from đ??´ to planes passing through đ?›ź. Solution to Problem 230

231. Consider a plane đ?›ź that passes through the midpoint of segment [đ??´đ??ľ]. Show that points đ??´ and đ??ľ are equally distant from plane đ?›ź. Solution to Problem 231

232. Through a given point, draw a line that intersects a given line and is ⊼ to another given line. Solution to Problem 232

233. Let đ?›ź and đ?›˝ be two distinct planes and the line đ?‘‘ their intersection. Let đ?‘€ be a point that is not located on đ?›ź âˆŞ đ?›˝. We draw the lines đ?‘€đ?‘€1 and đ?‘€đ?‘€2 ⊼ on đ?›ź and đ?›˝. Show that the line đ?‘‘ is ⊼ to (đ?‘€đ?‘€1 đ?‘€đ?‘€2 ). Solution to Problem 233

234. A plane đ?›ź and a point đ??´, đ??´ ∉ đ?›ź are given. Find the locus of points đ?‘€ ∈ đ?›ź such that segment |đ??´đ?‘€| has a given length. Solution to Problem 234

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255 Compiled and Solved Problems in Geometry and Trigonometry

235. Let đ?‘‚, đ??´, đ??ľ, đ??ś be four points such that đ?‘‚đ??´ ⊼ đ?‘‚đ??ľ ⊼ đ?‘‚đ??ś ⊼ đ??ˇđ??´ and we write đ?‘Ž = ‖đ?‘‚đ??´â€–, đ?‘? = ‖đ?‘‚đ??ľâ€–, đ?‘? = ‖đ?‘‚đ??śâ€–. a. Find the length of the sides of ∆đ??´đ??ľđ??ś in relation to đ?‘Ž, đ?‘?, đ?‘?; b. Find đ?œŽ[đ??´đ??ľđ??ś] and demonstrate the relation đ?œŽ[đ??´đ??ľđ??ś]2 = đ?œŽ[đ??ˇđ??´đ??ľ]2 + đ?œŽ[đ?‘‚đ??ľđ??ś]2 + đ?œŽ[đ?‘‚đ??śđ??´]2 ;

c. Show that the orthogonal projection of point đ?‘‚ on plane (đ??´đ??ľđ??ś) is the orthocenter đ??ť of ∆đ??´đ??ľđ??ś; d. Find the distance ‖đ?‘‚đ??ťâ€–. Solution to Problem 235

236. Consider non-coplanar points đ??´, đ??ľ, đ??ś, đ??ˇ and lines đ??´đ??´â€˛, đ??ľđ??ľâ€˛, đ??śđ??śâ€˛, đ??ˇđ??ˇâ€˛ perpendicular to (đ??ľđ??śđ??ˇ), (đ??´đ??śđ??ˇ), (đ??´đ??ľđ??ˇ). Show that if lines đ??´đ??´â€˛ and đ??ľđ??ľâ€˛ are concurrent, then lines đ??śđ??śâ€˛, đ??ˇđ??ˇâ€˛ are coplanar. Solution to Problem 236

237. Let đ??´, đ??ľ, đ??ś, đ??ˇ four non-coplanar points. Show that đ??´đ??ľ ⊼ đ??śđ??ˇ and đ??´đ??ś ⊼ đ??ľđ??ˇ â&#x;š đ??´đ??ˇ ⊼ đ??ľđ??ś. Solution to Problem 237

238. On the edges of a triangle with its peak đ?‘‚, take the points đ??´, đ??ľ, đ??ś such that |đ?‘‚đ??´| ≥ |đ?‘‚đ??ľ| ≥ |đ?‘‚đ??ś|. Show that the ⊼ foot in đ?‘‚ to the plane (đ??´đ??ľđ??ś) coincides with the point of intersection of the bisectors ∆đ??´đ??ľđ??ś. Solution to Problem 238

239. Let a peak đ??´ of the isosceles triangle đ??´đ??ľđ??ś (|đ??´đ??ľ| ≥ |đ??´đ??ś|) be the orthogonal projection onto đ??´â€˛ on a plane đ?›ź which passes through đ??ľđ??ś. Show ′ đ??ś > đ??ľđ??´đ??ś Ě‚ Ě‚. that đ??ľđ??´

Solution to Problem 239

240. With the notes of Theorem 1, let [đ??´đ??ľ ′ be the opposite ray to [đ??´đ??ľ ′′ . Show ′′ đ??´đ??ľ > đ?‘€đ??´đ??ľ Ě‚. that for any point đ?‘€ ∈ đ?›źâ€“ [đ??´đ??ľ ′′ we have đ??ľĚ‚

Solution to Problem 240 179

Florentin Smarandache

241. Let đ?›ź be a plane, đ??´ ∈ đ?›ź and đ??ľ and đ??ś two points on the same side of đ?›ź Ě‚ is the complement of the angle formed such that đ??´đ??ś ⊼ đ?›ź. Show that đ??śđ??´đ??ľ by [đ??´đ??ľ with đ?›ź. Solution to Problem 241

242. Let đ?›źâ€˛đ?›˝â€˛ be a trihedral angle with edge đ?‘š and đ??´ ∈ đ?‘š. Show that of all the rays with origin at đ??´ and contained in half-plane đ?›˝â€˛, the one that forms with plane đ?›ź the biggest possible angle is that ⊼ đ?‘? ∈ đ?‘š (its support is called the line with the largest slope of đ?›˝ in relation to đ?›ź). Solution to Problem 242

243. Let đ?›ź be a plane, đ?œŽ a closed half-plane, bordered by đ?›ź, đ?›ź ′ a half-plane contained in đ?›ź and đ?‘Ž a real number between 00 and 1800 . Show that there is only one half-space đ?›˝ ′ that has common border with đ?›ź ′ such that đ?›˝ ′ ⊂ đ?œŽ and đ?‘š(đ?›ź ′ đ?›˝ ′ ) = đ?‘Ž. Solution to Problem 243

′ � ′ ) be a proper dihedral angle. Construct a half-plane � ′ such that ̂ 244. Let (� ′ � ′ ) = �(� ′ � ′ ). Show that the problem has two solutions, one of which ̂ ̂ �(� ′ � ′ (called bisector half-plane of � ′ � ′ ). ̂ ̂ is located in the int. �

Solution to Problem 244

245. Show that the locus of the points equally distant from two secant planes �, � is formed by two ⊼ planes, namely by the union of the bisector planes of the dihedral angles � and �. Solution to Problem 245

246. If đ?›ź and đ?›˝ are two planes, đ?‘„ ∈ đ?›˝ and đ?‘‘ ⊼ through đ?‘„ on đ?›ź. Show that đ?‘‘ ⊂ đ?›˝. Solution to Problem 246

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255 Compiled and Solved Problems in Geometry and Trigonometry

247. Consider a line � ⊂ �. Show that the union of the ⊼ lines to �, which intersect line �, is a plane ⊼ �. Solution to Problem 247

248. Find the locus of the points equally distant from two concurrent lines. Solution to Problem 248

249. Show that a plane � ⊼ to two secant planes is ⊼ to their intersection. Solution to Problem 249

250. Let đ??´ be a point that is not on plane đ?›ź. Find the intersection of all the planes that contain point đ??´ and are ⊼ to plane đ?›ź. Solution to Problem 250

251. From a given point draw a ⊼ plane to two given planes. Solution to Problem 251

252. Intersect a dihedral angle with a plane as the angle of sections is right. Solution to Problem 252

253. Show that a line � and a plane �, which are perpendicular to another plane, are parallel or line � is contained in �. Solution to Problem 253

254. If three planes are ⊼ to a plane, they intersect two by two after lines đ?‘Ž, đ?‘?, đ?‘?. Show that đ?‘Ž âˆĽ đ?‘? âˆĽ đ?‘?. Solution to Problem 254

255. From a point đ??´ we draw perpendicular lines đ??´đ??ľ and đ??´đ??ś to the planes of ′ đ?›˝ ′ . Show that đ?‘š(đ??ľđ??´đ??ś ′ đ?›˝ ′ ) or Ě‚ Ě‚ Ě‚ ) = đ?‘š(đ?›ź the faces of a dihedral angle đ?›ź ′ đ?›˝ ′ ). Ě‚ Ě‚ ) = 1800 − đ?‘š(đ?›ź đ?‘š(đ??ľđ??´đ??ś

Solution to Problem 255 181

Florentin Smarandache

Solutions Solution to Problem 201. đ?‘€1 − đ?‘§1 = 1 đ?‘€2 − đ?‘§1 = 2 + đ?‘– đ?‘€1 − đ?‘§1 = đ?‘Ľ + đ?‘Śđ?‘– ∆đ?‘€1 đ?‘€2 đ?‘€3 equilateral â&#x;š ‖đ?‘€1 đ?‘€2 ‖ = ‖đ?‘€1 đ?‘€3 ‖ = ‖đ?‘€2 đ?‘€3 ‖ ⇒ |đ?‘§2 − đ?‘§1 | = |đ?‘§3 − đ?‘§2 | = |đ?‘§1 − đ?‘§3 | ⇒ √2 = √(đ?‘Ľ − 2)2 + (đ?‘Ś − 1)2 ⇒ {

đ?‘Ľ+đ?‘Ś =2 (đ?‘Ľ − 2)2 + (đ?‘Ś − 1)2 = 2 ⇒{ 2 2 2 đ?‘Ľ + đ?‘Ś 2 − 2đ?‘Ľ = 1 (1 − đ?‘Ľ) + đ?‘Ś = 2

⇒ đ?‘Ś = 2−đ?‘Ľ đ?‘Ľ 2 + 4 + đ?‘Ľ 2 − 4đ?‘Ľ − 2đ?‘Ľ = 1 ⇒ đ?‘Ľ1,2 3+√3 1−√3

Thus: đ?‘€3 (

2

,

2

1 − √3 đ?‘Ś1 = 3 Âą √3 2 = ⇒ 2 1 + √3 đ?‘Ś 2 [ 2

3−√3 1+√3

) or đ?‘€3 (

2

,

2

). There are two solutions!

Solution to Problem 202. đ?‘§1 = đ?‘&#x;(cos đ?‘Ą1 + đ?‘– sin đ?‘Ą1 ) đ?‘§2 = đ?‘&#x;(cos đ?‘Ą2 + đ?‘– sin đ?‘Ą2 ) đ?‘§3 = đ?‘&#x;(cos đ?‘Ą3 + đ?‘– sin đ?‘Ą3 ) đ?‘§1 ≠đ?‘§2 ≠đ?‘§3 ⇒ đ?‘Ą1 ≠đ?‘Ą2 ≠đ?‘Ą3 đ?‘§1 + đ?‘§2 đ?‘§3 ∈ â„? ⇒ sin đ?‘Ą1 + đ?‘&#x; sin(đ?‘Ą2 + đ?‘Ą3 ) = 0 {đ?‘§2 + đ?‘§3 đ?‘§1 ∈ â„? ⇒ sin đ?‘Ą2 + đ?‘&#x; sin(đ?‘Ą1 + đ?‘Ą3 ) = 0 â&#x;š đ?‘§3 + đ?‘§1 đ?‘§2 ∈ â„? ⇒ sin đ?‘Ą3 + đ?‘&#x; sin(đ?‘Ą1 + đ?‘Ą2 ) = 0 sin đ?‘Ą1 (1 − đ?‘&#x; cos đ?‘Ą) + đ?‘&#x; sin đ?‘Ą ∙ cos đ?‘Ą1 = 0 {sin đ?‘Ą2 (1 − đ?‘&#x; cos đ?‘Ą) + đ?‘&#x; sin đ?‘Ą ∙ cos đ?‘Ą2 = 0 sin đ?‘Ą3 (1 − đ?‘&#x; cos đ?‘Ą) + đ?‘&#x; sin đ?‘Ą ∙ cos đ?‘Ą3 = 0 đ?‘Ą1 ≠đ?‘Ą2 ≠đ?‘Ą3 These equalities are simultaneously true only if 1 − đ?‘&#x; ∙ cos đ?‘Ą = 0 and đ?‘&#x; ∙ sin đ?‘Ą = 0, as đ?‘&#x; ≠0 ⇒ sin đ?‘Ą = 0 ⇒ đ?‘Ą = 0 ⇒ cos đ?‘Ą = 1 ⇒ 1 − đ?‘&#x; = 0 ⇒ đ?‘&#x; = 1, so đ?‘§1 đ?‘§2 đ?‘§3 = 1 ∙ (cos 0 + sin 0) = 1.

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255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 203. a.

𝜀𝑘 =

So

2𝑘𝜋 + 𝑛

2𝑖𝜋 𝑛 2𝑗𝜋 cos 𝑛

𝜀𝑖 = cos 𝜀𝑗 =

𝑖 sin

2𝑘𝜋 𝑛

, 𝑘 ∈ {0, 1, … , 𝑛 − 1}.

2𝑖𝜋 𝑛 2𝑗𝜋} + 𝑖 sin 𝑛

+ 𝑖 sin

⟹ 𝜀𝑖 𝜀𝑗 = cos

2𝜋(𝑖+𝑗) 𝑛

+ 𝑖 sin

2𝜋(𝑖+𝑗) , 𝑖, 𝑗 𝑛

∈ {0, 1, … , 𝑛 − 1}.

1) 𝑖 + 𝑗 < 𝑛 − 1 ⟹ 𝑖 + 𝑗 = 𝑘 ∈ {0, 1, … , 𝑛 − 1} ⟹ 𝜀𝑖 𝜀𝑗 = 𝜀𝑘 ∈ 𝐺; 2) 𝑖 + 𝑗 = 𝑛 ⟹ 𝜀𝑖 𝜀𝑗 = cos 2𝜋 + 𝑖 sin 2𝜋 = 1 = 𝜀𝑜 ∈ 𝐺; 2𝜋(𝑛∙𝑚+𝑟) 2𝜋(𝑛∙𝑚+𝑟) + 𝑖 sin 𝑛 𝑛 2𝜋𝑟 𝑖 sin = 𝜀𝑟 ∈ 𝐺. 𝑛

3) 𝑖 + 𝑗 > 𝑛 ⟹ 𝑖 + 𝑗 = 𝑛 ∙ 𝑚 + 𝑟, 0 ≤ 𝑟 < 𝑛, 𝜀𝑖 𝜀𝑗 = cos 2𝜋𝑟 2𝜋𝑟 cos (2𝜋𝑚 + ) + 𝑖 sin (2𝜋𝑚 + ) 𝑛 𝑛 2𝜋𝑖 2𝜋𝑖 𝜀𝑖 = cos + 𝑖 sin 𝑛 𝑛

b.

1

𝜀𝑖−1 = 𝜀 = 𝑖

𝑖 sin (2𝜋 −

cos 0+𝑖 sin 0 2𝜋𝑖 2𝜋𝑖 +𝑖 sin 𝑛 𝑛

cos

2𝜋𝑖 ) 𝑛

= cos

= cos (−

2𝜋𝑛−2𝜋𝑖 𝑛

2𝜋𝑖 ) 𝑛

+ 𝑖 sin

=

2𝜋𝑟 cos 𝑛

+ 𝑖 sin (−

2𝜋𝑛−2𝜋𝑖 𝑛

= cos

+

2𝜋𝑖 ) 𝑛

= cos (2𝜋 −

2𝜋(𝑛−1) + 𝑛

𝑖 sin

=

2𝜋𝑖 )+ 𝑛

2𝜋(𝑛−1) 𝑛

,

𝑖 ∈ {0, 1, … , 𝑛 − 1}.

If 𝑖 = 0 ⟹ 𝑛 − 𝑖 = 𝑛 ⟹ 𝜀0−1 = 𝜀0 ∈ 𝑔. If 𝑖 ≠ 0 ⟹ 𝑛 − 𝑖 ≤ 𝑛 − 1 ⟹ ℎ = 𝑛 − 𝑖 ∈ {0, 1, … , 𝑛 − 1} ⟹ 𝜖−1 = cos 𝑖 sin

2𝜋ℎ 𝑛

2𝜋ℎ 𝑛

+

∈ 𝐺.

Solution to Problem 204. 𝑎 = 𝑟1 (cos 𝑡1 + 𝑖 sin 𝑡1 ) {𝑏 = 𝑟2 (cos 𝑡2 + 𝑖 sin 𝑡2 ) 𝑐 = 𝑟3 (cos 𝑡3 + 𝑖 sin 𝑡3 ) arg𝑎 + arg𝑐 = 2arg𝑏 ⟹ 𝑡1 + 𝑡3 = 2𝑡2 and |𝑎| + |𝑐| = |𝑏| ⟹ 𝑟1 + 𝑟3 = 𝑟2 −𝑏 ± √𝑏 2 − 4𝑎𝑐 2𝑎 2 −𝑟2 (cos 𝑡2 + 𝑖 sin 𝑡2 ) ± √𝑟2 (cos 2𝑡2 + 𝑖 sin 2𝑡2 ) − 4𝑟1 𝑟3 (cos(𝑡1 + 𝑡3 ) + 𝑖 sin(𝑡1 + 𝑡3 )) = 2𝑟1 (cos 𝑡1 + 𝑖 sin 𝑡1 ) 𝑎𝑧 2 + 𝑏𝑧 + 𝑐 = 0 ⟹ 𝑧1,2 =

=

−𝑟2 (cos 𝑡2 + 𝑖 sin 𝑡2 ) ± √(cos 2𝑡2 + 𝑖 sin 2𝑡2 )(𝑟22 − 4𝑟1 𝑟3 ) 2𝑟1 (cos 𝑡1 + 𝑖 sin 𝑡1 )

But 𝑟1 + 𝑟3 = 𝑟2 ⟹ 𝑟22 = 𝑟12 + 𝑟12 + 𝑟32 + 2𝑟1 𝑟3 ⟹ 𝑟22 − 4𝑟1 𝑟3 = 𝑟12 + 𝑟12 + 𝑟32 + 2𝑟1 𝑟3 − 4𝑟1 𝑟3 = (𝑟1 − 𝑟3 )2. 183

Florentin Smarandache

Therefore: �1,2 =

−đ?‘&#x;2 (cos đ?‘Ą2 + đ?‘– sin đ?‘Ą2 ) Âą (cos đ?‘Ą2 + đ?‘– sin đ?‘Ą2 )(đ?‘&#x;1 − đ?‘&#x;3 ) 2đ?‘&#x;1 (cos đ?‘Ą1 + đ?‘– sin đ?‘Ą1 )

We observe that: �2 =

(cos đ?‘Ą2 +đ?‘– sin đ?‘Ą2 )(−2đ?‘&#x;1 ) 2đ?‘&#x;1 (cos đ?‘Ą1 +đ?‘– sin đ?‘Ą1 )

= −[cos(đ?‘Ą2 − đ?‘Ą1 ) + đ?‘– sin(đ?‘Ą2 − đ?‘Ą1 )] = cos[đ?œ‹ + (đ?‘Ą2 − đ?‘Ą1 )] +

đ?‘– sin[đ?œ‹ + đ?‘Ą2 − đ?‘Ą1 ] and đ?‘Ą2 = 1.

Solution to Problem 205. Let

Let

So � is determined.

So đ?›˝ is determined. If we work with reduced arguments, then đ?‘Ą4 = đ?‘Ą2 − đ?‘Ą1 or đ?‘Ą4 = đ?‘Ą2 − đ?‘Ą1 + 2đ?œ‹, in the same way đ?‘Ą5 .

184

255 Compiled and Solved Problems in Geometry and Trigonometry

According to a. (∃) the complex numbers of modulus 1, đ?›ź and đ?›˝ such that đ?‘§2 = đ?›źđ?‘§1 and đ?‘§3 = đ?›˝đ?‘§1 . In the given relation, by substitution we obtain:

� = 1 and � = 1 verify this equality, so in this case �2 = �3 = �1 . According to point b.,

where đ?›˝ = đ?‘Ľ + đ?‘–đ?‘Ś, when

We construct the system:

The initial solution leads us to �1 = �2 = �3 .

and gives

By substituting,

185

Florentin Smarandache

|�| = 2 does not comply with the condition |�| = 1. But

so

If then

and then

If are on the circle with radius đ?‘&#x; and the arguments are

they are the peaks of an equilateral triangle.

Solution to Problem 206. a. We assume that đ?‘‘ ∊ đ?›ź ≠∅ and đ?‘‘′ ∊ đ?›ź = {đ??ľ}.

186

255 Compiled and Solved Problems in Geometry and Trigonometry

Let đ?‘‘ ∊ đ?›ź = {đ??´} and đ?‘‘′ ∊ đ?›ź = {đ??ľ} and the planes determined by pairs of concurrent lines (đ?‘‘, đ??´đ??ľ); (đ?‘‘′, đ??´đ??ľ). We remark that these are the required planes, because

b. We assume đ?‘‘ ∊ đ?›ź = {đ??´} and đ?‘‘′||đ?›ź. We draw through đ??´, in plane đ?›ź, line đ?‘‘′||đ?‘‘ and we consider planes (đ?‘‘, đ?‘‘′′) and (đ?‘‘′, đ?‘‘′′) and we remark that

c. We assume đ?‘‘ ∊ đ?›ź = ∅ and đ?‘‘′ ∊ đ?›ź = ∅ and đ?‘‘′ ∈ direction đ?‘‘. Let đ??´ ∈ đ?›ź and đ?‘‘′′||đ?‘‘ â&#x;š đ?‘‘′′||đ?‘‘′ and the planes are (đ?‘‘, đ?‘‘′′) and (đ?‘‘′, đ?‘‘′′). The reasoning is the same as above.

Solution to Problem 207.

187

Florentin Smarandache

Solution to Problem 208.

If đ?›ź separates points đ??´ and đ??ľ, it means they are in different half-spaces and let đ?œŽ = |đ?›źđ??´ and đ?œŽâ€˛ = |đ?›źđ??ľ. Because đ?›ź separates đ??´ and đ??ś â&#x;š đ??ś ∈ đ?œŽâ€˛. Because đ?›ź separates đ??ś and đ??ˇ â&#x;š đ??ˇ ∈ đ?œŽ. From đ??ľ ∈ đ?œŽâ€˛ and đ??ˇ ∈ đ?œŽ â&#x;š đ?›ź separates points đ??ľ and đ??ˇ

From đ??´ ∈ đ?œŽ and đ??ˇ ∈ đ?œŽ â&#x;š |đ??ľđ??ˇ| ∊ đ?›ź = ∅.

Solution to Problem 209.

188

255 Compiled and Solved Problems in Geometry and Trigonometry

From (1), (2), (3)

Solution to Problem 210.

Ě‚ , |đ?‘‰đ??ś) = |(đ?‘‰đ??´đ??ľ), đ??ś ∊ |(đ?‘‰đ??ľđ??ś), đ??´ ∊ |(đ?‘‰đ??´đ??ś), đ??ľ is thus an intersection of a. int. (|đ?‘‰đ??´, |đ?‘‰đ??ľ convex set and thus the interior of a trihedron is a convex set. b. Tetrahedron [đ?‘‰đ??´đ??ľđ??ś] without edge [đ??´đ??ś]. We mark with â„ł1 = [đ??´đ??ľđ??ś]– [đ??´đ??ś] = [đ??´đ??ľ, đ??ś ∊ [đ??ľđ??ś, đ??´ ∊ |đ??´đ??ś, đ??ľ is thus a convex set, being intersection of convex sets. is a convex set. In the same way is a convex set, where But [đ?‘‰đ??´đ??ľđ??ś] – [đ??´đ??ś] = and thus it is a convex set as intersection of convex sets. c. Tetrahedron [đ?‘‰đ??´đ??ľđ??ś] without face [đ??´đ??ľđ??ś] đ?‘‰ is thus intersection of convex sets â&#x;š is a convex set. 189

Florentin Smarandache

Solution to Problem 211.

In plane (đ??ľđ??´đ??ś) we have đ??¸đ??š||đ??´đ??ś. In plane (đ??ˇđ??´đ??ś) we have đ??´đ??ś ⊂ (đ??ˇđ??´đ??ś) â&#x;š đ??¸đ??š||(đ??ˇđ??´đ??ś). In this plane we also have đ??ťđ??ş||đ??´đ??ś. So đ??¸đ??š||đ??ťđ??ş â&#x;š đ??¸, đ??š, đ??ş, đ??ť are coplanar and because ‖đ??¸đ??šâ€– =

‖đ??´đ??śâ€– 2

= ‖đ??ťđ??şâ€– â&#x;š đ??¸đ??šđ??şđ??ť is a parallelogram.

Solution to Problem 212. We assume we have đ?›ź||đ?›˝||đ?›ž such that đ??´đ??´â€™ ⊂ đ?›ź, đ??ľđ??ľâ€™ ⊂ , đ??śđ??śâ€™ ⊂ đ?›ž.

We draw through đ??´â€˛ a parallel line with đ?‘‘: đ?‘‘′′||đ?‘‘. As đ?‘‘ intersects all the 3 planes đ??´â€˛ ⊂ đ?‘‘′′ at đ??´, đ??ľ, đ??ś â&#x;š and its || đ?‘‘’’ cuts them at đ??´â€˛, đ??ľâ€˛â€˛, đ??śâ€˛â€˛. Because

Let plane (đ?‘‘′ , đ?‘‘′′ ). Because this plane has in common with planes đ?›ź, đ?›˝, đ?›ž the points đ??´â€˛, đ??ľâ€˛â€˛, đ??śâ€˛â€˛ and because đ?›ź || đ?›˝ || đ?›ž â&#x;š it intersects them after the parallel lines

Taking into consideration (1) and (2)

The vice-versa can be similarly proved.

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255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 213.

Let

such that

and

such that

So

according to problem 7, three planes can be drawn || � || � || � such that

and đ?‘ƒđ?‘ƒâ€˛ ⊂ đ?›ź. So by marking đ?‘ƒ and letting đ?‘ƒâ€˛ variable, đ?‘ƒâ€˛ ∈ a parallel plane with the two lines, which passes through đ?‘ƒ. It is known that this plane is unique, because by drawing through đ?‘ƒ parallel lines to đ?‘‘ and đ?‘‘’ in order to obtain this plane, it is well determined by 2 concurrent lines.

Vice-versa: Let đ?‘ƒ ∈ đ?›ź, that is the plane passing through đ?‘ƒ and it is parallel to đ?‘‘ and đ?‘‘’. 191

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(đ?‘ƒâ€˛â€˛, đ?‘‘) determines a plane, and (đ?‘ƒâ€˛â€˛, đ?‘‘′) determines a plane â&#x;š the two planes, which have a common point, intersect after a line(đ?‘ƒâ€˛â€˛, đ?‘‘)∊ (đ?‘ƒâ€˛â€˛, đ?‘‘′) = đ?‘„đ?‘„′ where đ?‘„ ∈ đ?‘‘ and đ?‘„′ ∈ đ?‘‘′. Because such that đ?‘€đ?‘„ ⊂ đ?›˝, đ?›˝||đ?›ź. Because such that đ?‘€â€˛đ?‘„′ ⊂ đ?›ž, đ?›ž||đ?›ź. So the required locus is a parallel plane with đ?‘‘ and đ?‘‘’.

Solution to Problem 214.

We consider the plane, which according to a previous problem, represents the locus of the points dividing the segments with extremities on lines đ?‘‘1 and đ?‘‘3 in a given ratio đ?‘˜. To obtain this plane, we take a point đ??´ ∈ đ?‘‘1 , đ??ľ ∈ đ?‘‘3 and point đ??ś ∈ đ??´đ??ľ such that

‖đ??´đ??śâ€– ‖đ??śđ??ľâ€–

= đ?‘˜. Through this point đ??ś we draw two parallel lines đ?‘‘1 and đ?‘‘3 which

determine the above mentioned plane đ?›ź. Let đ?‘‘2 ∊ đ?›ź = {đ?‘ }. We must determine a segment that passes through đ?‘ and with its extremities on đ?‘‘1 and đ?‘‘3 , respectively at đ?‘€ and đ?‘ƒ. As the required line passes through đ?‘ and đ?‘€

The same line must pass through đ?‘ and đ?‘ƒ and because

đ?‘€, đ?‘ , đ?‘ƒ collinear.

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255 Compiled and Solved Problems in Geometry and Trigonometry

From (1) and (2) Then, according to previous problem 8:

and the required line is đ?‘€đ?‘ƒ.

Solution to Problem 215.

Let ∆đ?‘€đ?‘ đ?‘ƒ such that

Let ∆đ?‘€â€˛đ?‘ ′đ?‘ƒâ€˛ such that

Line đ?‘€đ?‘ƒ generates a plane đ?›˝, being parallel to a fixed direction đ?‘‘1 and it is based on a given line đ?‘‘. In the same way, the line đ?‘€đ?‘ generates a plane đ?›ž, parallel to a fixed direction đ?‘‘2 , and based on a given line đ?‘‘. As đ?‘‘ is contained by đ?›ž â&#x;š đ?‘‚ is a common point for đ?›ź and đ?›ž â&#x;š đ?›ź ∊ đ?›ž ≠∅ â&#x;š đ?›ź ∊ đ?›ž = đ?‘‘′, đ?‘‚ ∈ đ?‘‘′.

(∀) the considered ∆, so đ?‘ also describes a line đ?‘‘′ ⊂ đ?›ź. Because plane đ?›ž is well determined by line đ?‘‘ and direction đ?‘‘2 , is fixed, so đ?‘‘′ = đ?›ź ∊ đ?›ž is fixed. In the same way, đ?‘ƒđ?‘ will generate a plane đ?›ż, moving parallel to the fixed direction đ?‘‘3 and being based on the given line đ?‘‘’. As 193

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(∀)đ?‘ƒ variable peak, đ?‘ƒ ∈ đ?‘‘′′. Thus, in the given conditions, for any ∆đ?‘€đ?‘ đ?‘ƒ, peak đ?‘ƒ ∈ đ?‘‘′′. Vice-versa, let đ?‘ƒâ€˛ ∈ đ?‘‘′′. On plane (đ?‘‘′, đ?‘‘′′) we draw đ?‘ƒâ€˛, đ?‘€â€˛||đ?‘ƒđ?‘€ â&#x;š (đ?‘€â€˛đ?‘ƒâ€˛đ?‘ ′)||(đ?‘ƒđ?‘€đ?‘ ) â&#x;š (đ?‘‘đ?‘‘′) the intersection of two parallel planes after parallel lines đ?‘€â€˛đ?‘ ′||đ?‘€đ?‘ and the so constructed ∆đ?‘€â€˛đ?‘ƒâ€˛đ?‘ ′ has its sides parallel to the three fixed lines, has đ?‘€â€˛ ∈ đ?‘‘ and đ?‘ ′ ∈ đ?›ź, so it is one of the triangles given in the text. So the locus is line đ?‘‘′′. We’ve seen how it can be constructed and it passes through đ?‘‚. In the situation when đ??ˇ||đ?›ź we obtain

In this case the locus is a parallel line with đ?‘‘.

Let đ?‘€đ?‘ đ?‘ƒ and đ?‘€â€˛đ?‘ ′đ?‘ƒâ€˛ such that

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255 Compiled and Solved Problems in Geometry and Trigonometry

We assume đ?›ź ∊ đ?›˝ = đ?‘‘ and let đ?‘‘ ∊ (đ?‘€đ?‘ đ?‘ƒ) = {đ?‘‚} and đ?‘‘ ∊ (đ?‘€â€˛đ?‘ ′đ?‘ƒâ€˛) = {đ?‘‚′} â&#x;š

a plane cuts the parallel planes after parallel lines. In the same way, đ?‘‚đ?‘ ||đ?‘‚′đ?‘ ′ and because

We use the property: Let đ?œ‹1 and đ?œ‹2 2 parallel planes and đ??´, đ??ľ, đ??ś ⊂ đ?œ‹1 and đ??´â€˛đ??ľâ€˛đ??śâ€˛ ⊂ đ?œ‹2 , đ??´đ??ľ âˆĽ đ??´â€˛đ??ľâ€˛,

Let’s show that đ??ľđ??ś||đ??ľâ€˛đ??śâ€˛. Indeed (đ??ľđ??ľâ€˛đ??śâ€˛) is a plane which intersects the 2 planes after parallel lines.

Applying in (1) this property â&#x;š đ?‘‚đ?‘ƒ||đ?‘‚′đ?‘ƒâ€˛. Maintaining đ?‘‚đ?‘ƒ fixed and letting đ?‘ƒâ€™ variable, always đ?‘‚đ?‘ƒ||đ?‘‚′đ?‘ƒâ€˛.=, so đ?‘‚′đ?‘ƒâ€˛ generates a plane which passes through đ?‘‘. We assume đ?›˝||đ?›ź.

đ?‘€đ?‘ đ?‘ ′đ?‘€â€˛ parallelogram

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Considering đ?‘ƒâ€˛ fix and đ?‘ƒ variable â&#x;š đ?‘ƒđ?‘ƒâ€˛||đ?›ź and the set of parallel lines drawn đ?‘ƒđ?‘ƒâ€˛||đ?›˝ to a plane through an exterior point is a parallel plane with the given plane. So the locus is a parallel plane with đ?›ź and đ?›˝.

Solution to Problem 216.

In plane đ??ˇđ??´đ??ś we have:

In plane đ??ˇđ??´đ??ľ we have:

In plane đ?‘‚đ??ľđ??ś we have:

From đ?‘ƒđ?‘€||đ??´đ??ś and đ?‘ƒđ?‘ ||đ??ľđ??ś â&#x;š (đ?‘€đ?‘ đ?‘…)||(đ??´đ??ľđ??ś). Let đ?‘„ and đ??ˇ be midpoints of sides |đ?‘€đ?‘ | and |đ??´đ??ľ|.

are collinear. 196

255 Compiled and Solved Problems in Geometry and Trigonometry

Concurrent lines đ?‘‚đ??ˇ and đ?‘‚đ??ś determine a plane which cuts the parallel planes

are collinear. So đ??şâ€™ ∈ |đ?‘‚đ??ş â&#x;š the required locus is ray |đ?‘‚đ??ş.

Vice-versa: we take a point on |đ?‘‚đ??ş, đ??şâ€˛â€˛, and draw through it a parallel plane to (đ??´đ??ľđ??ś), plane (đ?‘€â€˛â€˛, đ?‘ ′′, đ?‘ƒâ€˛â€˛), similar triangles are formed and the ratios from the hypothesis appear. Solution to Problem 217. Let đ??´2 , đ??ľ2 , đ??ś2 , đ??ˇ2 such that

Mark on lines đ??´đ??ˇ1 and đ??ľđ??ś1 points đ?‘€ and đ?‘ such that

From

Next is

The same,

As 197

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we obtain

From

is a parallelogram. is parallelogram. So is a parallelogram.

Solution to Problem 218.

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We draw through đ??´â€˛ a line đ?‘‘′′||đ?‘‘. We draw two parallel planes with đ?›ź, which will intersect the three lines in đ??ľâ€˛, đ??ľâˆ— , đ??ľ and đ??śâ€˛, đ??ś ∗ , đ??ś. Plane (đ?‘‘, đ?‘‘∗ ) intersects planes đ?›ź, (đ??ľâ€˛đ??ľâˆ— đ??ľ), (đ??śâ€˛đ??ś ∗ đ??ś) after parallel lines

= ‖đ??ľđ??ľâˆ— ‖ = ‖đ??śâ€˛đ??ś ∗ ‖. Plane (đ?‘‘′, đ?‘‘∗ ) intersects parallel planes (đ??ľâ€˛đ??ľâˆ— đ??ľ), (đ??śâ€˛đ??ś ∗ đ??ś) after parallel lines

So (∀) parallel plane with đ?›ź we construct, the newly obtained triangle has a side of ∗ đ??ľ is constant. We mark with a line Ě‚ đ?›ź length and the corresponding angle to đ??ľâ€˛đ??ľ

that position of the plane, for which the opposite length of the required angle is đ?‘™. With the compass spike at đ??ś and with a radius equal with đ?‘™, we trace a circle arc that cuts segment | đ??śâ€˛đ??ś ∗ | at đ?‘ or line đ??śâ€˛đ??ś ∗. Through đ?‘ we draw at (đ?‘‘′, đ?‘‘∗ ) a parallel line to đ?‘‘∗ which precisely meets đ?‘‘′ in a point đ?‘€â€˛. Through đ?‘€â€˛, we draw the || plane to đ?›ź, which will intersect the three lines in đ?‘€, đ?‘€â€˛, đ?‘€âˆ— .

is a parallelogram.

â&#x;š đ??śđ?‘ đ?‘€â€˛đ?‘€ is a parallelogram. and line đ?‘€đ?‘€â€˛, located in a parallel plane to đ?›ź, is parallel to đ?›ź.

Discussion: Ě‚âˆ— are constant, then Assuming the plane (đ??śâ€˛đ??ś ∗ đ??ś) is variable, as | đ??śđ??ś ∗ | and đ??śđ??śâ€˛đ??ś đ?‘‘(đ??śâ€˛đ??ś ∗ đ??ś) = đ?‘? = also constant If đ?‘™ < đ?‘‘ we don’t have any solution. If đ?‘™ = đ?‘‘ (∃) a solution, the circle of radius đ?‘™, is tangent to đ??śâ€˛đ??ś ∗. If đ?‘™ > đ?‘‘ (∃ ) two solutions: circle of radius đ?‘™, cuts đ??śâ€˛đ??ś ∗ at two points đ?‘ and đ?‘ƒ.

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Solution to Problem 219.

We draw through đ??´ planes đ?›ź ⊼ đ?‘‘ and đ?›źâ€˛ ⊼ đ?‘‘′. As đ??´ is a common point â&#x;š đ?›ź ∊ đ?›ź ′ = ∆â&#x;š đ??´ ∈ ∆.

â&#x;š ∆ the required line If đ?›ź ≠đ?›źâ€˛ - we have only one solution. If đ?›ź = đ?›źâ€˛ (∀) line from đ?›ź which passes through đ??´ corresponds to the problem, so (∃) infinite solutions.

Solution to Problem 220.

Let đ?‘‘1 ⊼ đ?‘‘2 two concurrent perpendicular lines, đ?‘‘1 ∊ đ?‘‘2 = {đ?‘‚}. They determine a plane đ?›ź = (đ?‘‘1 , đ?‘‘2 ) and đ?‘‚ ∈ đ?›ź. We construct on đ?›ź in đ?‘‚.

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255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 221. We use the reductio ad absurdum method. Let đ?‘‘ ⊼ đ?‘Ž, đ?‘‘ ⊼ đ?‘Ž, đ?‘‘ ⊼ đ?‘?. We assume that these lines are not coplanar. Let đ?›ź = (đ?‘?, đ?‘?), đ?›źâ€˛ = (đ?‘Ž, đ?‘?), đ?›ź ≠đ?›źâ€˛. Then đ?‘‘ ⊼ đ?›ź, đ?‘‘ ⊼ đ?›źâ€™. Thus through point đ?‘‚, 2 perpendicular planes to đ?‘‘ can be drawn. False â&#x;š đ?‘Ž, đ?‘?, đ?‘? are coplanar.

Solution to Problem 222. By reductio ad absurdum: Let đ?‘Ž ∊ đ?‘? ∊ đ?‘? ∊ đ?‘‘ = {đ?‘‚} and they are perpendicular two by two. From đ?‘‘ ⊼ đ?‘Ž, đ?‘‘ ⊼ đ?‘Ž, đ?‘‘ ⊼ đ?‘?

â&#x;š đ?‘Ž, đ?‘?, đ?‘? are coplanar and đ?‘? ⊼ đ?‘Ž, đ?‘? ⊼ đ?‘Ž, so we can draw to point đ?‘‚ two

distinct perpendicular lines. False. So the 4 lines cannot be perpendicular two by two.

Solution to Problem 223.

We assume that đ?‘‘ ⊼ đ?›ź. In đ?‘‘′ ∊ đ?›ź = {đ?‘‚} we draw line đ?‘‘′′ ⊼ đ?›ź. Lines đ?‘‘′ and đ?‘‘′′ are concurrent and determine a plane đ?›˝ = (đ?‘‘′, đ?‘‘′′) and as đ?‘‚′ ∈ đ?›˝, đ?‘‚′ ∈ đ?›ź â&#x;š

From (1) and (2) â&#x;š in plane đ?›˝, on line đ?‘Ž, at point đ?‘‚′ two distinct perpendicular lines had been drawn. False. So đ?‘‘′||đ?›ź.

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Solution to Problem 224.

Reductio ad absurdum. Let đ?‘‘ âˆŚ đ?‘‘′. We draw đ?‘‘′′||đ?‘‘ through đ?‘‚’.

â&#x;š at point đ?‘‚′ we can draw two perpendicular lines to plane đ?›ź. False. So đ?‘‘||đ?‘‘′.

Solution to Problem 225.

Let đ?‘‘ ⊼ đ?›ź and đ?‘‘ ∊ đ?›ź = {đ?‘‚}. We draw through đ?‘‚ a parallel to đ?‘‘′, which will be contained in đ?›ź, then đ?‘‘||đ?›ź.

Solution to Problem 226.

We assume đ?›˝ â&#x;š đ?›ź ∊ đ?›˝ ≠∅ and let đ??´ ∈ đ?›ź ∊ đ?›˝ â&#x;š through a point đ??´ there can be drawn two distinct perpendicular planes on this line. False. â&#x;š đ?›ź || đ?›˝. 202

255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 227.

Let đ?‘€ be a point in space with the property ||đ?‘€đ??´|| = ||đ?‘€đ??ľ||. We connect đ?‘€ with the midpoint of segment [đ??´đ??ľ], point đ?‘‚.

So đ?‘€ is on a line drawn through đ?‘‚, perpendicular to đ??´đ??ľ. But the union of all perpendicular lines drawn through đ?‘‚ to đ??´đ??ľ is the perpendicular plane to đ??´đ??ľ at point đ?‘‚, marked with đ?›ź, so đ?‘€ ∈ đ?›ź.

Vice-versa: let đ?‘€ ∈ đ?›ź,

common side

Solution to Problem 228.

Let đ?‘€ be a point in space with this property:

Let đ?‘‚ be the center of the circumscribed circle ∆đ??´đ??ľđ??ś â&#x;š ||đ?‘‚đ??´|| = đ?‘‚đ??ľ|| = ||đ?‘‚đ??ś||, so đ?‘‚ is also a point of the desired locus.

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According to the previous problem the locus of the points in space equally distant from đ??´ and đ??ľ is in the mediator plane of segment [đ??´đ??ľ], which also contains đ?‘€. We mark with đ?›ź this plane. The locus of the points in space equally distant from đ??ľ and đ??ś is in the mediator plane of segment [đ??ľđ??ś], marked đ?›˝, which contains both đ?‘‚ and đ?‘€. So đ?›ź ∊ đ?›˝ = đ?‘‚đ?‘€.

so đ?‘€ ∈ the perpendicular line to plane (đ??´đ??ľđ??ś) in the center of the circumscribed circle ∆đ??´đ??ľđ??ś.

Vice-versa, let đ?‘€ ∈ this perpendicular line

= ||đ??śđ?‘€||, so đ?‘€ has the property from the statement.

Solution to Problem 229.

We draw ⊼ from đ??ľ to the plane. Let đ?‘‚ be the foot of this perpendicular line. Let

the circle of radius đ?‘‚đ??´. Vice-versa, let đ?‘€ ∈ this circle

so đ?‘€ represents the foot from đ??ľ to đ??´đ?‘€.

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255 Compiled and Solved Problems in Geometry and Trigonometry

Solution to Problem 230. Let đ?›ź be a plane that passes through đ?‘Ž and let đ?‘€ be the ⊼ foot from đ??´ to đ?›ź â&#x;š đ??´đ?‘€ ⊼ đ?›ź. From

so đ?‘€ ∈ a perpendicular line to đ?‘Ž in đ??´â€˛, thus it is an element of the perpendicular plane to đ?‘Ž in đ??´â€˛, which we mark as đ?œ‹ and which also contains đ??´. đ?‘€ ∈ the circle of radius đ??´đ??´â€˛ from plane Ď€.

*Vice-versa, let đ?‘€ be a point on this circle of radius đ??´đ??´â€˛ from plane đ?œ‹.

â&#x;š đ?‘€ is the foot of a ⊼ drawn from đ??´ to a plane that passes through đ?‘Ž.

Solution to Problem 231.

Let đ??´â€˛ and đ??ľâ€˛ be the feet of the perpendicular lines from đ??´ and đ??ľ to đ?›ź

(∃) a plane đ?›˝ = (đ??´đ??´â€˛, đ??ľđ??ľâ€˛) and đ??´đ??ľ ⊂ đ?›˝ 205

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are collinear. In plane đ?›˝ we have

right

Solution to Problem 232.

Let đ?‘‘, đ?‘‘′ be given lines, đ??´ given point. We draw through đ??´ plane đ?›ź ⊼ to đ?‘‘′. If đ?‘Ž ∊ đ?›ź = {đ??ľ}, then line đ??´đ??ľ is the desired one, because it passes through đ??´, meets đ?‘‘ and from đ?‘‘′ đ?›ź đ?‘‘′ đ??´đ??ľ. If đ?‘‘ ∊ đ?›ź = ∅ there is no solution. If đ?‘‘ ⊂ đ?›ź, then any line determined by đ??´ and a point of đ?‘‘ represents solution to the problem, so there are infinite solutions.

Solution to Problem 233.

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Solution to Problem 234.

Let đ?‘€ be a point such that ||đ??´đ?‘€|| = đ?‘˜. We draw đ??´đ??´â€˛ ⊼ đ?›ź ⇒ đ??´â€˛ fixed point and đ??´đ??´â€˛ ⊼ đ??´â€˛đ?‘€. We write ||đ??´đ??´â€˛|| = đ?‘Ž. Then

đ?‘€ ∈ a circle centered at đ??´â€˛ and of radius √đ?‘˜ 2 − đ?‘Ž2 , for đ?‘˜ > đ?‘Ž. For đ?‘˜ = đ?‘Ž we obtain 1 point. For đ?‘˜ < đ?‘Ž empty set. Vice-versa, let đ?‘€ be a point on this circle â&#x;š

so đ?‘€ has the property from the statement.

Solution to Problem 235.

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In ∆đ?‘‚đ??śđ?‘€:

But

3. Let đ??ť be the projection of đ?‘‚ lcp. plane đ??´đ??ľđ??ś, so

đ??ť ∈ corresponding heights of side đ??´đ??ľ. We show in the same way that đ??´đ??ś ⊼ đ??ľđ??ť and thus đ??ť is the point of intersection of the heights, thus orthocenter.

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Solution to Problem 236.

First we prove that if a line is ⊼ to two concurrent planes â&#x;š the planes coincide.

Let

∆đ??´đ??ľđ?‘€ has two right angles. False. We return to the given problem.

being concurrent, they determine a plane â&#x;š đ??śđ??ˇ ⊼ đ??´đ??ľ.

đ??ś, đ??ˇ, đ??śâ€˛, đ??ˇâ€˛ are coplanar â&#x;š đ??śđ??śâ€˛ and đ??ˇđ??ˇâ€˛ are coplanar. 209

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Solution to Problem 237.

We draw

â&#x;š đ??ľđ??´â€˛ height in ∆đ??ľđ??śđ??ˇ (1)

â&#x;š đ??´â€˛đ??ś height in ∆đ??´đ??ľđ??ś (2) From (1) and (2) â&#x;š đ??´â€˛ is the orthocenter ∆đ??´đ??ľđ??ś â&#x;š đ?‘‚đ??´â€˛ ⊼ đ??ľđ??ś.

Solution to Problem 238.

Let

∆đ??ľđ?‘‚đ?‘‚′ and ∆đ??śđ?‘‚đ?‘‚′ are right at đ?‘‚′.

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As

|đ?‘‚đ??´| ≥ |đ?‘‚đ??ľ| ≥ |đ?‘‚đ??ś| } |đ?‘‚đ?‘‚′| common side

is the center of the circumscribed circle ∆đ??´đ??ľđ??ś.

Solution to Problem 239.

Let đ??ˇ be the midpoint of [đ??ľđ??ś] and đ??¸ ∈ |đ??ˇđ??´â€˛ such that ||đ??ˇđ??¸|| = ||đ??ˇđ??´||. đ??´đ??ˇ is median in the ∆ isosceles

common

being external for

Solution to Problem 240.

Let đ?‘€ be a point in the plane and |đ??´đ?‘€â€˛ the opposite ray to đ??´đ?‘€. According to theorem 1

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Solution to Problem 241.

We construct đ??ľ on the plane

â&#x;š đ??´đ??ś and đ??ľđ??ľâ€˛ determine a plane đ?›˝ = (đ??´đ??ś, đ??ľđ??ľâ€˛) â&#x;š đ??´đ??ľ ⊂ đ?›˝ and on this plane Ě‚ . Ě‚ ) = 900 − đ?‘š(đ??ľđ??´đ??ľâ€˛) đ?‘€(đ??śđ??´đ??ľ

Solution to Problem 242.

Let ray |đ??´đ??ľ ⊂ đ?›˝â€˛ such that đ??´đ??ľ ⊼ đ?‘š. Let |đ??´đ??ś another ray such that |đ??´đ??ś ⊂ đ?›˝â€™. We draw Ě‚ > đ??śđ??´đ??śâ€˛ Ě‚. đ??ľđ??ľâ€˛ ⊼ đ?›ź and đ??śđ??śâ€˛ ⊼ đ?›ź to obtain the angle of the 2 rays with đ?›ź, namely đ??ľđ??´đ??ľâ€˛ We draw line |đ??´đ??´â€™ such that đ??´đ??´â€˛ ⊼ đ?›ź and is on the same side of plane đ?›ź as well as half-plane đ?›˝â€™.

[đ??´đ??ľ is the projection of ray [đ??´đ??´â€˛ on plane đ?›˝

Solution to Problem 243. Let đ?‘‘ be the border of đ?›źâ€˛ and đ??´ ∈ đ?‘‘. We draw a plane ⊼ on đ?‘‘ in đ??´, which we mark as đ?›ž.

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255 Compiled and Solved Problems in Geometry and Trigonometry

In this plane, there is only one ray đ?‘?, with its origin in đ??´, such that đ?‘š(đ?‘?,Ě‚đ?‘?) = đ?‘Ž. The desired half-plane is determined by đ?‘‘ and ray đ?‘?, because from

Solution to Problem 244.

Let đ?‘‘ be the edge of the dihedral angle and đ??´ ∈ đ?‘‘. We draw đ?‘Ž ⊼ đ?‘‘, đ?‘Ž ⊂ đ?›źâ€˛ and đ?‘? ⊼ đ?‘‘, đ?‘? ⊂ đ?›˝â€™ two rays with origin in đ??´. It results đ?‘‘ ⊼ (đ?‘Žđ?‘?). We draw on plane (đ?‘Ž, đ?‘?) ray đ?‘? Ě‚ ) (1). such that đ?‘š(đ?‘Žđ?‘? Ě‚) = đ?‘š(đ?‘?đ?‘? As đ?‘‘ ⊼ (đ?‘Žđ?‘?) â&#x;š đ?‘‘ ⊼ đ?‘?. Half-plane đ?›žâ€˛ = (đ?‘‘, đ?‘?) is the desired one, because

If we consider the opposite ray to đ?‘?, đ?‘?′, half-plane đ?›žâ€˛â€˛ = (đ?‘‘, đ?‘?′) also forms concurrent angles with the two half-planes, being supplementary to the others.

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Solution to Problem 245. Let đ?‘€ be a point in space equally distant from the half-planes đ?›źâ€˛, đ?›˝â€˛ â&#x;š ||đ?‘€đ??´|| = ||đ?‘€đ??ľ||.

where đ?‘‘ = đ?›ź ∊ đ?›˝. Let

|đ?‘€đ??´| = |đ?‘€đ??ľ| }â&#x;š |đ?‘‚đ?‘€| common side right triangle â&#x;š đ?‘€ ∈ bisector of the angle. Ě‚ â&#x;š đ?‘€ ∈ bisector half-plane of the angle of half-planes đ?›źâ€˛, đ?›˝â€˛. đ??´đ??ˇđ??ľ If đ?‘€â€˛ is equally distant from half-planes đ?›˝â€˛ and đ?›źâ€˛â€˛ we will show in the same way that đ?‘€â€˛ ∈ bisector half-plane of these half-planes. We assume that đ?‘€ and đ?‘€â€˛ are on Ě‚ ) = 900 , so the two half-planes are ⊼. this plane ⊼ to đ?‘‘, we remark that đ?‘š(đ?‘€đ?‘‚đ?‘€â€˛ Considering the two other dihedral angles, we obtain 2 perpendicular planes, the 2 bisector planes.

Vice-versa: we can easily show that a point on these planes is equally distant form planes � and �.

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Solution to Problem 246.

Let đ?›ź ∊ đ?›˝ = đ?›ź. In plane đ?›˝ we draw

As

but

so from a point it can be drawn only one perpendicular line to a plane,

Solution to Problem 247.

Let

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the line with the same direction. We know that the union of the lines with the same direction and are based on a given line is a plane. As this plane contains a perpendicular line to �, it is perpendicular to �.

Solution to Problem 248.

Let � = (�1 , �2 ) the plane of the two concurrent lines and � is a point with the property �(�, �1 ) = �(�, �2 ). We draw Let

a bisector of the angle formed by the two lines, and đ?‘€ is on a line đ?›ź which meets a bisector ⇒ đ?‘€ ∈ a plane ⊼ đ?›ź and which intersects đ?›ź after a bisector. Thus the locus will be formed by two planes ⊼ Îą and which intersects đ?›ź after the two bisectors of the angle formed by đ?‘‘1 , đ?‘‘2 . The two planes are ⊼. ‖đ?‘€â€˛đ??´â€– = ‖đ?‘€â€˛đ??ľâ€– â&#x;š } â&#x;š đ?‘€đ??´ ⊼ đ?‘‘1 ||MM′|| common side And in the same way đ?‘€đ??ľ ⊼ đ?‘‘2 ⇒ đ?‘€ has the property from the statement.

Solution to Problem 249.

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255 Compiled and Solved Problems in Geometry and Trigonometry

Let đ?›˝ ∊ đ?›ž = đ?‘‘ and đ?‘€ ∈ đ?‘‘ â&#x;š đ?‘€ ∈ đ?›˝, đ?‘€ ∈ đ?›ž. We draw ⊼ from đ?‘€ to đ?›ź, line đ?‘‘’. According to a previous problem

Solution to Problem 250.

Let � and � be such planes, that is

From

are secant planes and ⊼ to đ?›ź. So their intersection is ⊼ through đ??´ to plane đ?›ź.

Solution to Problem 251. We construct the point on the two planes and the desired plane is determined by the two perpendicular lines.

Solution to Problem 252. Let đ?›ź ∊ đ?›˝ = đ?‘‘ and đ?‘€ ∈ đ?‘‘. We consider a ray originating in đ?‘€, đ?‘Ž ∈ đ?›ź and we construct a ⊼ plane to đ?‘Ž in đ?‘€, plane đ?›ž. Because

217

Florentin Smarandache

and let a ray originating in đ?‘€, đ?‘? ⊂ đ?›˝ ∊ đ?›ž ⇒ đ?‘? ⊂ đ?›˝ , đ?‘? ⊂.đ?›ž As đ?‘Ž ⊼ đ?›ž â&#x;š đ?‘Ž ⊼ đ?‘? and the desired plane is that determined by rays (đ?‘Ž, đ?‘?).

Solution to Problem 253.

Let đ?›ź ∊ đ?›˝ = đ?‘Ž and đ?‘‘ ∊ đ?›˝ = {đ??´}. We suppose that đ??´ ∉ đ?‘Ž. Let đ?‘€ ∈ đ?‘Ž, we build đ?‘? ⊼ đ?›˝, đ?‘€ ∈ đ?‘? â&#x;š đ?‘? ⊂ đ?›ź. If

Solution to Problem 254.

218

255 Compiled and Solved Problems in Geometry and Trigonometry

From (1), (2), (3) ⟹ 𝑎 ∥ 𝑏 ∥ 𝑐.

Solution to Problem 255.

′ 𝛽 ′ ), 𝛼 ∩ 𝛽 = 𝑑. ̂ Let 𝐴 ∈ int. (𝛼

′′ 𝛽 ′ ). We show the same way that Let 𝐴 ∈ int. (𝛼̂ ′′ 𝛽 ′ ) ̂ ) = 1800 − 𝑚(𝛼̂ 𝑚(𝐵𝐴𝐶 ′ 𝛽 ′ ) = 𝑚(𝛼 ′ 𝛽 ′ ). ̂ ̂ ̂ ) = 1800 − 1800 + 𝑚(𝛼 } ⟹ 𝑚(𝐵𝐴𝐶 ′′ 𝛽 ′ ) = 1800 − 𝑚(𝛼 ′ 𝛽′ ) ̂ 𝑚(𝛼̂ ′′ 𝛽 ′′ ) ⟹ 𝑚(𝐵𝐴𝐶 ′′ 𝛽 ′ ). ̂ ) = 1800 − 𝑚(𝛼̂ If 𝐴 ∈ int. (𝛼̂ ′ 𝛽 ′′ ) ⟹ 𝑚(𝐵𝐴𝐶 ′ 𝛽 ′ ). ̂ ̂ ) = 1800 − 𝑚(𝛼 If 𝐴 ∈ int. (𝛼̂

219

Florentin Smarandache

This book is a translation from Romanian of "Probleme Compilate Ĺ&#x;i Rezolvate de Geometrie Ĺ&#x;i Trigonometrie" (University of Kishinev Press, Kishinev, 169 p., 1998), and includes problems of 2D and 3D Euclidean geometry plus trigonometry, compiled and solved from the Romanian Textbooks for 9th and 10th grade students, in the period 1981-1988, when I was a professor of mathematics at the "Petrache Poenaru" National College in Balcesti, Valcea (Romania), LycĂŠe Sidi El Hassan Lyoussi in Sefrou (Morroco), then at the "Nicolae Balcescu" National College in Craiova and Dragotesti General School (Romania), but also I did intensive private tutoring for students preparing their university entrance examination. After that, I have escaped in Turkey in September 1988 and lived in a political refugee camp in Istanbul and Ankara, and in March 1990 I immigrated to United States. The degree of difficulties of the problems is from easy and medium to hard. The solutions of the problems are at the end of each chapter. One can navigate back and forth from the text of the problem to its solution using bookmarks. The book is especially a didactical material for the mathematical students and instructors. The Author

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