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HMMT 二月 2008 · TEAM2 赛 · 第 12 题

HMMT February 2008 — TEAM2 Round — Problem 12

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 35 ] Show that the incenter of triangle AEF lies on the incircle of ABC .
解析
  1. [ 35 ] Show that the incenter of triangle AEF lies on the incircle of ABC . Solution: Let segment AI meet the incircle at A . Let us show that A is the incenter of 1 1 AEF . 5 A A 1 F E I B C D ′ Since AE = AF and AA is the angle bisector of ∠ EAF , we find that A E = A F . Using 1 1 tangent-chord, we see that ∠ AF A = ∠ A EF = ∠ A F E . Therefore, A lies on the angle 1 1 1 1 bisector of ∠ AF E . Since A also lies on the angle bisector of ∠ EAF , A must be the incenter 1 1 of AEF , as desired.