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HMMT 二月 2011 · TEAM2 赛 · 第 10 题

HMMT February 2011 — TEAM2 Round — Problem 10

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 15 ] Let the circumcircle of triangle AOM intersect ω again at D . Prove that points A , D , and X arecollinear.
解析

英文解析

  1. [ 15 ] Let the circumcircle of triangle AOM intersect ω again at D . Prove that points A , D , and X arecollinear.
    Solution: By the similarity 4 OAM ∼ OXA , we have that ∠ OAX = ∠ OM A . Since AOM D is acyclic quadrilateral, we have that ∠ OM A = ∠ ODA . Since OA = OD , we have that ∠ ODA = ∠ OAD .
    Combining these equations tells us that ∠ OAX = ∠ OAD , so A , D , and X are collinear, as desired.