HMMT 二月 2011 · 冲刺赛 · 第 1 题

HMMT February 2011 — Guts Round — Problem 1

[ 4 ] Let ABC be a triangle with area 1. Let points D and E lie on AB and AC , respectively, such that DE is parallel to BC and DE/BC = 1 / 3. If F is the reflection of A across DE , find the area of triangle F BC .

解析

[ 4 ] Let ABC be a triangle with area 1. Let points D and E lie on AB and AC , respectively, such that DE is parallel to BC and DE/BC = 1 / 3. If F is the reflection of A across DE , find the area of triangle F BC . 1 Answer: 3 Let AF intersect BC at H . Since DE/BC = 1 / 3 and F and A are equidistant from DE , we have 2 1 AF = AH and F H = AH − AF = AH . Furthermore, since AF is perpendicular to DE , we have 3 3 AH and F H are the altitudes of triangles ABC and F BC respectively. Therefore the area of triangle 1 1 1 1 F BC is · F H · BC = · · AH · BC = . 2 2 3 3