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HMMT 二月 2008 · TEAM2 赛 · 第 11 题

HMMT February 2008 — TEAM2 Round — Problem 11

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 15 ] Show that lines AD, BE, CF pass through a common point.
解析

英文解析

  1. [ 15 ] Show that lines AD, BE, CF pass through a common point.
    Solution: Using Ceva’s theorem on triangle ABC , we see that it suffices to show that
    BD CE AF
    · · = 1 .
    DC EA F B
    Since AF = AE , BD = BF , and CD = CE (due to equal tangents), we see that the LHS isindeed 1.
    Remark: The point of concurrency is known as the Gergonne point .