HMMT 二月 2008 · TEAM2 赛 · 第 12 题

英文解析

12. [ 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 .
    A5
    1A
    B CIEF
    ′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.