February, 2017

Geometry Difficulty 4.8 AIME Prove it United States

Problem:

Let $ABCD$ be a convex quadrilateral with $AC = 7$ and $BD = 17$ . Let $M, P, N, Q$ be the midpoints of sides $AB$ , $BC$ , $CD$ , $DA$ respectively. Compute $MN^{2} + PQ^{2}$ .

Solution

Solution:

$MPNQ$ is a parallelogram whose side lengths are $3.5$ and $8.5$ so the sum of squares of its diagonals is $\frac{7^{2} + 17^{2}}{2} = 169$

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.