HMMT 十一月 2016 · GEN 赛 · 第 10 题

HMMT November 2016 — GEN Round — Problem 10

Quadrilateral ABCD satisfies AB = 8, BC = 5, CD = 17, DA = 10. Let E be the intersection of AC and BD . Suppose BE : ED = 1 : 2. Find the area of ABCD .

解析

Quadrilateral ABCD satisfies AB = 8, BC = 5, CD = 17, DA = 10. Let E be the intersection of AC and BD . Suppose BE : ED = 1 : 2. Find the area of ABCD . Proposed by: Brice Huang Answer: 60 Since BE : ED = 1 : 2, we have [ ABC ] : [ ACD ] = 1 : 2. Suppose we cut off triangle ACD , reflect it across the perpendicular bisector of AC , and re-attach it ′ ′ ′ ′ ′ as triangle A C D (so A = C, C = A ). ′ ′ ′ ′ ′ ′ Triangles ABC and C A D have vertex A = C and bases BC and A D . Their areas and bases are ′ ′ both in the ratio 1 : 2. Thus in fact BC and A D are collinear. ′ ′ ′ ′ 1 Hence the union of ABC and C A D is the 8 − 15 − 17 triangle ABD , which has area · 8 · 15 = 60. 2