Scientia Magna Vol. 6 (2010), No. 1, 7-8
Smarandache’s ratio theorem Edited by M. Khoshnevisan Neuro Intelligence Center, Australia E-mail: mkhoshnevisan@neurointelligence-center.org Abstract In this paper we present the Smarandache’s ratio theorem in the geometry of the triangle. Keywords Smarandache’s ratio theorem, triangle.
§1. The main result Smarandache’s Ratio Theorem If the points A1 , B1 , C1 divide the sides k BC k= a, k CA k= b, respectively k AB k= c of a triangle 4ABC in the same ratio k > 0, then k AA1 k2 + k BB1 k2 + k CC1 k2 ≥
3 2 (a + b2 + c2 ). 4
Proof. Suppose k > 0 because we work with distances. k BA1 k= k k BC k, k CB1 k= k k CA k, k AC1 k= k k AB k . We’ll apply tree times Stewart’s theorem in the triangle 4ABC, with the segments AA1 , BB1 , respectively CC1 : k AB k2 · k BC k (1 − k)+ k AC k2 · k BC k k− k AA1 k2 · k BC k=k BC k3 (1 − k)k, where k AA1 k2 = (1 − k) k AB k2 +k k AC k2 −(1 − k)k k BC k2 . Similarly, k BB1 k2 = (1 − k) k BC k2 +k k BA k2 −(1 − k)k k AC k2 . k CC1 k2 = (1 − k) k CA k2 +k k CB k2 −(1 − k)k k AB k2 . By adding these three equalities we obtain: k AA1 k2 + k BB1 k2 + k CC1 k2 = (k 2 − k + 1)(k AB k2 + k BC k2 + k CA k2 ), 1 which takes the minimum value when k = , which is the case when the three lines from the 2 enouncement are the medians of the triangle. 3 The minimum is (k AB k2 + k BC k2 + k CA k2 ). 4
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Edited by M. Khoshnevisan
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§2. Open problems on Smarandache’s ratio theorem 1. If the points A01 , A02 , . . . , A0n divide the sides A1 A2 , A2 A3 , . . . , An A1 of a polygon in a ratio k > 0, determine the minimum of the expression: k A1 A01 k2 + k A2 A02 k2 + · · · + k An A0n k2 . 2. Similarly question if the points A01 , A02 , . . . , A0n divide the sides A1 A2 , A2 A3 , . . . , An A1 in the positive ratios k1 , k2 , . . . , kn respectively. 3. Generalize this problem for polyhedrons.
References [1] F. Smarandache, Probl` emes avec et san. . . probl` emes!, Problem 5. 38, p. 56, Somipress, F´ es, Morocco, 1983. [2] F. Smarandache, Eight Solved and Eight Open Problems in Elementary Geometry, in arXiv.org, Cornell University, NY, USA. [3] M. Khoshnevisan, Smarandache’s Ratio Theorem, NeuroIntelligence Center, Australia, http://www.scribd.com/doc/28317750/Smaradache-s-Ratio-Theorem.