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HMMT 二月 2017 · 几何 · 第 1 题

HMMT February 2017 — Geometry — Problem 1

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. Let A, B, C, D be four points on a circle in that order. Also, AB = 3, BC = 5, CD = 6, and DA = 4.
    Let diagonals AC and BD intersect at P . Compute .APCP
解析

英文解析

  1. Let A, B, C, D be four points on a circle in that order. Also, AB = 3, BC = 5, CD = 6, and DA = 4.
    Let diagonals AC and BD intersect at P . Compute .AP
    Proposed by: Sam Korsky cP
    Answer:2
    AP DP CP DP5
    Note that 4 AP B ∼ 4 DP C so = . Similarly, 4 BP C ∼ 4 AP D so = . Dividing these
    AB CD BC DAtwo equations yields
    AP AB · DA 2 = =
    CP BC · CD 5