PUMaC 2014 · 几何（A 组） · 第 6 题

PUMaC 2014 — Geometry (Division A) — Problem 6

[ 6 ] 4 ABC has side lengths AB = 15, BC = 34, and CA = 35. Let the circumcenter of ABC be O . Let D be the foot of perpendicular from C to AB . Let R be the foot of perpendicular from D to AC , and let W be the perpendicular foot from D to BC . Find the area of quadrilateral CROW .

解析

[ 6 ] 4 ABC has side lengths AB = 15, BC = 34, and CA = 35. Let the circumcenter of ABC be O . Let D be the foot of perpendicular from C to AB . Let R be the foot of perpendicular from D to AC , and let W be the perpendicular foot from D to BC . Find the area of quadrilateral CROW . Solution: By Heron, (area of 4 ABC ) = 252. Let P be the perpendicular foot from O to AC , and let Q be the perpendicular foot from O to BC . Then, it’s clear that 4 DOP = 4 W OP and 4 ABC 4 DOQ = 4 W OQ . From this, ( area of CROW ) = ( area of DP CQ ) = = 126 . 2