HMMT 二月 2017 · 几何 · 第 6 题

HMMT February 2017 — Geometry — Problem 6

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 AF meets M C at G , and line GD meets γ again at E . Determine the area of pentagon ABCDE .

解析

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 AF meets 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.