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HMMT 二月 2017 · 几何 · 第 6 题

HMMT February 2017 — 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. In convex quadrilateral ABCD we have AB = 15, BC = 16, CD = 12, DA = 25, and BD = 20. Let
    M and γ denote the circumcenter and circumcircle of 4 ABD . Line CB meets γ again at F , line AFmeets M C at G , and line GD meets γ again at E . Determine the area of pentagon ABCDE .
解析

英文解析

  1. In convex quadrilateral ABCD we have AB = 15, BC = 16, CD = 12, DA = 25, and BD = 20. Let
    M and γ denote the circumcenter and circumcircle of 4 ABD . Line CB meets γ again at F , line AFmeets M C at G , and line GD meets γ again at E . Determine the area of pentagon ABCDE .
    Proposed by: Evan Chen
    Answer: 396
    °
    Note that ∠ ADB = ∠ DCB = 90 and BC ‖ AD . Now by Pascal theorem on DDEBF A implies that
    B , M , E are collinear. So [ ADE ] = [ ABD ] = 150 and [ BCD ] = 96, so the total area is 396.