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HMMT 二月 2008 · TEAM1 赛 · 第 12 题

HMMT February 2008 — TEAM1 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. [ 40 ] Let K be as in the previous problem. Let M be the midpoint of BC and N the midpointof AC . Show that K lies on line M N .
解析

英文解析

  1. [ 40 ] Let K be as in the previous problem. Let M be the midpoint of BC and N the midpointof AC . Show that K lies on line M N .
    ° ′
    Solution: Since I, K, E, C are concyclic, we have ∠ IKC = ∠ IEC = 90 . Let C be the
    ′ ′
    reflection of C across BI , then C must lie on AB . Then, K is the midpoint of CC . Consider
    1 ′
    a dilation centered at C with factor . Since C lies on AB , it follows that K lies on M N .
    A2
    ′
    IKNC
    B C M