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HMMT 二月 2008 · 几何 · 第 6 题

HMMT February 2008 — Geometry — Problem 6

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  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
  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 √ √ BC Answer: 17 Using extended Sine law, we find the circumradius of ABC to be R = = 4 2. 2 sin A √ 2 2 2 By considering the power of point P , we find that R − OP = P B · P C = 15. So OP = R − 15 = √ √ 16 · 2 − 15 = 17.