Track / Stage 5 / 380 of 400 #980 of 1964

Problem 980

AIME late

Geometry Difficulty 6.0 Prove it

[ $\quad$ Properties and characteristics of a parallelogram $]$ [ Rectangles and squares. Properties and characteristics]

In parallelogram $A B C D$ , heights $B M$ and $B N$ are drawn from vertex $B$ of the obtuse angle, and heights $D P$ and $D Q$ are drawn from vertex $D$ .

Prove that points $M, N, P$ , and $Q$ are the vertices of a rectangle.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Let, for definiteness, point $N$ lie on line $A D$ , and point $Q$ - on line $A B$ (see fig.). Then the diagonals $B D$ and $P N$ of rectangle $P B N D$ are equal and intersect at their common midpoint $O$ . Similarly, the diagonals $B D$ and $Q M$ of rectangle $Q B M D$ are equal and intersect at their common midpoint $O$ . Therefore, the diagonals $P N$ and $Q M$ of quadrilateral $P Q N M$ are equal and intersect at their common midpoint $O$ . Hence, $P Q N M$ is a rectangle.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.