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HMMT 十一月 2015 · 冲刺赛 · 第 14 题

HMMT November 2015 — Guts Round — Problem 14

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 9 ] Let ABC be an acute triangle with orthocenter H . Let D, E be the feet of the A, B -altitudesrespectively. Given that AH = 20 and HD = 15 and BE = 56, find the length of BH .
解析

英文解析

  1. [ 9 ] Let ABC be an acute triangle with orthocenter H . Let D, E be the feet of the A, B -altitudesrespectively. Given that AH = 20 and HD = 15 and BE = 56, find the length of BH .
    Proposed by: Sam Korsky
    Answer: 50
    Let x be the length of BH . Note that quadrilateral ABDE is cyclic, so by Power of a Point, x (56 − x ) =
    20 · 15 = 300. Solving for x , we get x = 50 or 6. We must have BH > HD so x = 50 is the correctlength.