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HMMT 二月 2008 · 几何 · 第 6 题

HMMT February 2008 — Geometry — Problem 6

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 5 ] Let ABC be a triangle with ∠ A = 45 . Let P be a point on side BC with P B = 3 and P C = 5.
    Let O be the circumcenter of ABC . Determine the length OP .
解析

英文解析

  1. [ 5 ] Let ABC be a triangle with ∠ A = 45 . Let P be a point on side BC with P B = 3 and P C = 5.
    Let O be the circumcenter of ABC . Determine the length OP .
    √ √2
    Answer: 17 Using extended Sine law, we find the circumradius of ABC to be R = = 4 2.BC
    2 sin A
    √
    2 2
    By considering the power of point P , we find that R − OP = P B · P C = 15. So OP = R − 15 =2
    √
    √
    16 · 2 − 15 = 17.