Track / Stage 5 / 320 of 400 #920 of 1964

Problem 920

AIME late

Geometry Difficulty 5.8 Prove it

1.2. Prove that the midpoints of the sides of any quadrilateral are the vertices of a parallelogram. For which quadrilaterals is this parallelogram a rectangle, a rhombus, or a square?

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

1.2. Let $K, L, M$ and $N$ be the midpoints of the sides $A B, B C, C D$ and $D A$ of quadrilateral $A B C D$ , respectively. Then $K L = M N = A C / 2$ and segment $K L$ is parallel to $M N$ , i.e., $K L M N$ is a parallelogram. It is now clear that $K L M N$ is a rectangle if the diagonals $A C$ and $B D$ are perpendicular; a rhombus if $A C = B D$ ; a square if the diagonals $A C$ and $B D$ are equal in length and perpendicular.

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