HMMT 二月 2011 · TEAM2 赛 · 第 12 题
HMMT February 2011 — TEAM2 Round — Problem 12
题目详情
- [ 10 ] Prove that line AD is the symmedian from A in triangle ABC by showing that ∠ DAB = ∠ M AC .
解析
- [ 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 AD is the symmedian from A in triangle ABC , as desired.