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HMMT 二月 2008 · TEAM2 赛 · 第 13 题

HMMT February 2008 — TEAM2 Round — Problem 13

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 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

英文解析

  1. [ 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
    1A
    BIEF
    C1
    D1
    1 1 1
    ∠ DA C + ∠ B C A = ∠ DIC + ∠ B IF + ∠ F IA
    1 1 1 1 1 1 1 1
    2 2 2 = ( ∠ EID + ∠ DIF + ∠ F IE )1
    °14 = · 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