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HMMT 二月 2011 · TEAM2 赛 · 第 12 题

HMMT February 2011 — TEAM2 Round — Problem 12

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 10 ] Prove that line AD is the symmedian from A in triangle ABC by showing that ∠ DAB = ∠ M AC .
解析

英文解析

  1. [ 10 ] Prove that line AD is the symmedian from A in triangle ABC by showing that ∠ DAB = ∠ M AC .
    Solution: First, we have that
    1 π
    ∠ OAC = ( π − ∠ AOC ) = − ∠ ABC = ∠ HAB
    2 2
    so, by the previous problem, we obtain
    ∠ DAB = ∠ DAH + ∠ HAB = ∠ M AO + ∠ OAC = ∠ M AC .
    It follows that the lines AD and AM are reflections of each other across the bisector of ∠ BAC , so ADis the symmedian from A in triangle ABC , as desired.