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HMMT 二月 2007 · 几何 · 第 2 题

HMMT February 2007 — Geometry — Problem 2

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 3 ] A , B , C , and D are points on a circle, and segments AC and BD intersect at P , such that AP = 8,
    P C = 1, and BD = 6. Find BP , given that BP < DP.
解析

英文解析

  1. [ 3 ] A , B , C , and D are points on a circle, and segments AC and BD intersect at P , such that AP = 8,
    P C = 1, and BD = 6. Find BP , given that BP < DP.
    Answer: 2 . Writing BP = x and P D = 6 − x , we have that BP < 3. Power of a point at P gives
    AP · P C = BP · P D or 8 = x (6 − x ). This can be solved for x = 2 and x = 4, and we discard thelatter.