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HMMT 二月 2008 · 几何 · 第 4 题

HMMT February 2008 — Geometry — Problem 4

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 4 ] In a triangle ABC , take point D on BC such that DB = 14 , DA = 13 , DC = 4, and the circumcircleof ADB is congruent to the circumcircle of ADC . What is the area of triangle ABC ?
    ° °
解析

英文解析

  1. [ 4 ] In a triangle ABC , take point D on BC such that DB = 14 , DA = 13 , DC = 4, and the circumcircleof ADB is congruent to the circumcircle of ADC . What is the area of triangle ABC ?
    Answer: 108
    B CA
    M D
    The fact that the two circumcircles are congruent means that the chord AD must subtend the sameangle in both circles. That is, ∠ ABC = ∠ ACB , so ABC is isosceles. Drop the perpendicular M from
    A to BC ; we know M C = 9 and so M D = 5 and by Pythagoras on AM D , AM = 12. Therefore, the
    1 1
    area of ABC is ( AM )( BC ) = (12)(18) = 108.
    2 2