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HMMT 二月 2024 · 团队赛 · 第 3 题

HMMT February 2024 — Team Round — Problem 3

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [25] Let ABC be a scalene triangle and M be the midpoint of BC . Let X be the point such that ◦ CX ∥ AB and ∠ AM X = 90 . Prove that AM bisects ∠ BAX .
解析
  1. [25] Let ABC be a scalene triangle and M be the midpoint of BC . Let X be the point such that ◦ CX ∥ AB and ∠ AM X = 90 . Prove that AM bisects ∠ BAX . Proposed by: Pitchayut Saengrungkongka Solution: A X B C M Y Let Y be the intersection of lines AB and XM . Since BY ∥ CX , we have ∠ Y BM = ∠ XCM . Furthermore, we have BM = CM , since M is the midpoint of BC . Thus, ∼ △ BM Y △ CM X. = Thus, M Y = M X . Combined with the condition AM ⊥ XY , we get that AY X is an isosceles triangle with median AM . Therefore, AM bisects ∠ Y AX which is the same as ∠ BAX and we are done.