HMMT 二月 2017 · 团队赛 · 第 5 题

HMMT February 2017 — Team Round — Problem 5

[ 35 ] Let ABC be an acute triangle. The altitudes BE and CF intersect at the orthocenter H , and ◦ point O denotes the circumcenter. Point P is chosen so that ∠ AP H = ∠ OP E = 90 , and point Q is ◦ chosen so that ∠ AQH = ∠ OQF = 90 . Lines EP and F Q meet at point T . Prove that points A, T, O are collinear.

解析

[ 35 ] Let ABC be an acute triangle. The altitudes BE and CF intersect at the orthocenter H , and ◦ point O denotes the circumcenter. Point P is chosen so that ∠ AP H = ∠ OP E = 90 , and point Q is ◦ chosen so that ∠ AQH = ∠ OQF = 90 . Lines EP and F Q meet at point T . Prove that points A, T, O are collinear. Proposed by: Evan Chen Observe that T is the radical center of the circles with diameter OE , OF , AH . So T lies on the radical axis of ( OE ), ( OF ) which is the altitude from O to EF , hence passing through A . So AT O are collinear, done.