HMMT 二月 2008 · TEAM2 赛 · 第 13 题

HMMT February 2008 — TEAM2 Round — Problem 13

[ 35 ] Let A , B , C be the incenters of triangle AEF, BDF, CDE , respectively. Show that 1 1 1 A D, B E, C F all pass through the orthocenter of A B C . 1 1 1 1 1 1

解析

[ 35 ] Let A , B , C be the incenters of triangle AEF, BDF, CDE , respectively. Show that 1 1 1 A D, B E, C F all pass through the orthocenter of A B C . 1 1 1 1 1 1 Solution: Using the result from the previous problem, we see that A , B , C are respectively 1 1 1 the midpoints of the arc F E, F D, DF of the incircle. We have A 1 F E I B 1 C 1 D 1 1 1 ∠ DA C + ∠ B C A = ∠ DIC + ∠ B IF + ∠ F IA 1 1 1 1 1 1 1 1 2 2 2 1 = ( ∠ EID + ∠ DIF + ∠ F IE ) 4 1 ◦ = · 360 4 ◦ = 90 . It follows that A D is perpendicular to B C , and thus A D passes through the orthocenter 1 1 1 1 of A B C . Similarly, A D, B E, C F all pass through the orthocenter of A B C . 1 1 1 1 1 1 1 1 1