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HMMT 二月 2008 · TEAM2 赛 · 第 14 题

HMMT February 2008 — TEAM2 Round — Problem 14

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 X be the point on side BC such that BX = CD . Show that the excircle ABCopposite of vertex A touches segment BC at X .
解析

英文解析

  1. [ 40 ] Let X be the point on side BC such that BX = CD . Show that the excircle ABCopposite of vertex A touches segment BC at X .
    ′
    Solution: Let the excircle touch lines BC , AC and AB at X , Y and Z , respectively. Usingthe equal tangent property repeatedly, we have
    ′ ′
    BX − X C = BZ − CY = ( EY − CY ) − ( F Z − BZ ) = CE − BF = CD − BD.
    ′ ′
    It follows that BX = CD , and thus X = X . So the excircle touches BC at X .
    A6
    XEFT
    ZDCB Y