Selected Olympiad Geometry Problems 3 Russelle Guadalupe June 21, 2011
1. In an acute 4ABC with AC 6= BC, D, E are points on sides BC, AC respectively such that ABDE is cyclic. The diagonals AD and BE meet at P . Prove that if CP ⊥ AB, then P is the orthocenter of 4ABC. 2. Points A1 and C1 lie on sides AB and BC of the parallelogram ABCD respectively. Lines AC1 and CA1 meet at P . The circumcircles of triangles AA1 P and CC1 P intersect again at Q inside 4ACD. Prove that ∠P DA = ∠QBA.
3. In 4ABC with centroid G, D lies on ray AG different from A such that AG = GD and E lies on ray BG different from B such that BG = GE. M is the midpoint of AB. Prove that BM CD is cyclic iff BA = BE. 4. In 4ABC with incenter I, D, E are the midpoints of sides AC, AB respectively. DI meets AB at P and EI meets AC at Q. Prove that AP · AQ = AB · AC iff ∠A = 60◦ . 5. Let ABCD be a convex quadriateral with ∠BCD = ∠CDA. The internal bisector of ∠ABC intersects CD at E. Prove that ∠AEB = 90◦ iff AB = AD + AC. 6. Let I be the incenter of 4ABC and let γ be its circumcircle. The line AI meets γ again at D. Points \ and BC, respectively, such that E, F lie on arc BDC 1 ∠BAF = ∠CAE < ∠BAC. 2 Let G be the midpoint of IF . Prove that DG and EI intersect on γ.
1
7. Point D lies on side AC of 4ABC such that BD = CD. Point E lies on side BC, and the line through E parallel to BD meets AB at F . AE meets BD at G. Prove that ∠BCG = ∠BCF . 8. In an acute 4ABC with circumcenter O, CD is the altitude from C (with D ∈ AB), E is a point from side AB and M is the midpoint of CE. A line through M perpendicular to OM meets AC and BC at K and L, respectively. Prove that AD LM = . MK DB 9. Let ω be the circumcircle of the cyclic quadrilateral ABCD. The tangent lines to ω at B, C meet line AD at M, N respectively. BN intersects CM at E and AE intersects BC at F . If L is the midpoint of side BC, show that the circumcircle of 4DLF is tangent to ω at D.
10. In an acute 4ABC, D, E, F are the feet of the altitudes from A, B, C to sides BC, CA, AB respectively. A line through D parallel to EF intersects lines AC, AB at R, Q respectively. Lines BC and EF meet at P and M is the midpoint of BC. Prove that P, Q, M, R are concyclic. 11. In 4ABC, points D, E lie on sides AB, AC respectively. M, N, K are the midpoints of BE, CD, DE respectively. Prove that the circumcircle of 4M N K is tangent to DE iff OD = OE, where O is the circumcenter of 4ABC. 2
12. In a cyclic quadrilateral ABCD, denote X = AD ∩ BC, Y = AB ∩ CD, P = AC ∩ XY , Q = BD ∩ XY . Let E, F be the midpoints of AC, BD respectively. Prove that EF P Q is cyclic. 13. In a right 4ABC with ∠A = 90◦ and incenter I, BI meets AC at D and CI meets AB at E. Points P, Q lie on side BC such that IP k AB and IQ k AC. Prove that BE + CD = 2P Q. 14. In 4ABC, D lies on AC such that BD is the angle bisector of ∠ABC. Points P, Q lie on line AD such that AQ ⊥ BD and CP ⊥ BD. Let M be the midpoint of BC and O be the circumcenter of 4QM P . The circumcircle of 4QM P intersects AC again at H. Let E be the midpoint of BC. Prove that O, H, M, E are concyclic. 15. In 4ABC with orthocenter H, M is the midpoint of BC. Let D be the foot of the perpendicular from A to line P H. Prove that A, B, C, D are concyclic. 16. In 4ABC with circumcircle ω centered at O, the tangents to ω at B, C meet at T . A circle with center T and radius T B = T C meets the internal angle bisector of ∠BAC again at M . OM intesects BC at P , and E, F are the projections of M onto AC, AB respectively. Prove that P E ⊥ P F . 17. Let H be the orthocenter of an acute triangle ABC with circumcircle Γ. Let P be a point on the arc BC (not containing A) of Γ, and let M be a point on the arc CA (not containing B) of Γ such taht H lies on the segment P M . Let K be another point on Γ such that KM is parallel to the Simson line of P with respect to 4ABC. Let Q be another point on Γ such that P Q k BC. Segments BC and KQ intersect at point J. Prove that 4KJM is an isosceles triangle.
3