HMMT 二月 2003 · 冲刺赛 · 第 10 题

HMMT February 2003 — Guts Round — Problem 10

[7] Suppose that A, B, C, D are four points in the plane, and let Q, R, S, T, U, V be the respective midpoints of AB, AC, AD, BC, BD, CD . If QR = 2001 , SU = 2002 , T V = 2003, find the distance between the midpoints of QU and RV . 2 2 2 2 2

解析

Suppose that A, B, C, D are four points in the plane, and let Q, R, S, T, U, V be the respective midpoints of AB, AC, AD, BC, BD, CD . If QR = 2001 , SU = 2002 , T V = 2003, find the distance between the midpoints of QU and RV . Solution: 2001 This problem has far more information than necessary: QR and U V are both parallel to BC , and QU and RV are both parallel to AD . Hence, QU V R is a parallelogram, and the desired distance is simply the same as the side length QR , namely 2001. (See figure, next page.) 2 2 2 2 2