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HMMT 十一月 2012 · 冲刺赛 · 第 23 题

HMMT November 2012 — Guts Round — Problem 23

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 12 ] ABC is a triangle with AB = 4, BC = 6, and CA = 7. Let Ω be the circumcircle of ABC ; the tangent to Ω passing through A meets the extension of side BC at P . Find the length P B .
解析
  1. [ 12 ] 32 Answer: Since AP is a tangent to Ω, we know that ∠ P AB = ∠ P CA , so △ P AB ∼ △ P CA , so 11 we get that P B 4 P A = = . P A 7 P B + 6 Solving, we get that 7 P B = 4 P A , so 49 33 32 4( P B + 6) = 7 P A = P B ⇒ P B = 24 ⇒ P B = . 4 4 11