Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.0 AIME, harder Find the answer

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 $A B, A C, A D, B C, B D, C D$ . If $Q R=2001, S U=2002, T V=$ 2003, find the distance between the midpoints of $Q U$ and $R V$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

This problem has far more information than necessary: $Q R$ and $U V$ are both parallel to $B C$ , and $Q U$ and $R V$ are both parallel to $A D$ . Hence, $Q U V R$ is a parallelogram, and the desired distance is simply the same as the side length $Q R$ , namely 2001.

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