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HMMT 十一月 2012 · 冲刺赛 · 第 23 题

HMMT November 2012 — Guts Round — Problem 23

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 12 ] ABC is a triangle with AB = 4, BC = 6, and CA = 7. Let Ω be the circumcircle of ABC ; thetangent to Ω passing through A meets the extension of side BC at P . Find the length P B .
解析

英文解析

  1. [ 12 ]
    Answer: Since AP is a tangent to Ω, we know that ∠ P AB = ∠ P CA , so △ P AB ∼ △ P CA , so 32
    we get that 11
    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