PUMaC 2016 · 几何（A 组） · 第 7 题

PUMaC 2016 — Geometry (Division A) — Problem 7

Let ABCD be a cyclic quadrilateral with circumcircle ω and let AC and BD intersect at X . Let the line through A parallel to BD intersect line CD at E and ω at Y 6 = A . If AB = 10, AD = 24, XA = 17, and XB = 21, then the area of 4 DEY can be written in simplest form m as . Find m + n . n

解析

Let T be the foot of the altitude from A to BX . Observe that AT = 8 and BT = 6 (integer AT · EY Pythagorean triples). Then, [ DEY ] = ; it remains to find EY = AE − AY . 2 Observe that 4 DCX ∼ 4 ABX , and let r be the ratio of similarity; we can write CD = 10 r , DX = 17 r and CX = 21 r . Then: CA 21 r + 17 AE = XD · = 17 r · CX 21 r Since ABDY is an isosceles trapezoid, AY = BD − 2 BT = (21 + 17 r ) − 12 = 17 r + 9. Thus: 2 17 17 100 EY = AE − AY = (21 r + 17) − (17 r + 9) = − 9 = 21 21 21 1 100 400 Therefore, [ DEY ] = · 8 · = , so m + n = 421 . 2 21 21 Problem written by Eric Neyman. ′