HMMT 二月 2019 · 团队赛 · 第 6 题

HMMT February 2019 — Team Round — Problem 6

[ 45 ] Scalene triangle ABC satisfies ∠ A = 60 . Let the circumcenter of ABC be O, the orthocenter be H, and the incenter be I. Let D, T be the points where line BC intersects the internal and external angle bisectors of ∠ A, respectively. Choose point X on the circumcircle of 4 IHO such that HX ‖ AI. Prove that OD ⊥ T X.

解析

[ 45 ] Scalene triangle ABC satisfies ∠ A = 60 . Let the circumcenter of ABC be O, the orthocenter be H, and the incenter be I. Let D, T be the points where line BC intersects the internal and external angle bisectors of ∠ A, respectively. Choose point X on the circumcircle of 4 IHO such that HX ‖ AI. Prove that OD ⊥ T X. Proposed by: Wanlin Li ◦ Let I denote the A -excenter. Because ∠ A = 60 , AI is the perpendicular bisector of OH and A B, H, O, C all lie on the circle with diameter II . We are given that X is on this circle as well, and A since HI = OI, XIOI is also an isosceles trapezoid. But II is a diameter, so this means X must A A be diametrically opposite O on ( BOC ) and is actually the intersection of the tangents to ( ABC ) from B and C. Now ( T, D ; B, C ) = − 1 , so T is on the polar of D with respect to ( ABC ) . BC is the polar of X and D lies on BC, so X must also lie on the polar of D. Therefore T X is the polar of D with respect to ( ABC ) , and OD ⊥ T X as desired.