Track / Stage 5 / 384 of 400 #984 of 1964

Problem 984

AIME late

Geometry Difficulty 6.0 Prove it

14.5. Let $A B C D$ be a convex quadrilateral, and $K, L, M$ and $N$ be the midpoints of sides $A B, B C, C D$ and $D A$ . Prove that the intersection point of segments $K M$ and $L N$ is the midpoint of these segments, as well as the midpoint of the segment connecting the midpoints of the diagonals.

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

14.5. Place unit masses at the vertices of the quadrilateral $ABCD$ . Let $O$ be the center of mass of this system of points. It is sufficient to prove that point $O$ is the midpoint of segments $KM$ and $LN$ and the midpoint of the segment connecting the midpoints of the diagonals. Clearly, $K$ is the center of mass of points $A$ and $B$ , and $M$ is the center of mass of points $C$ and $D$ . Therefore, point $O$ is the center of mass of points $K$ and $M$ with masses 2, i.e., $O$ is the midpoint of segment $KM$ . Similarly, $O$ is the midpoint of segment $LN$ . Considering the centers of mass of pairs of points $(A, C)$ and $(B, D)$ (i.e., the midpoints of the diagonals), we obtain that point $O$ is the midpoint of the segment connecting the midpoints of the diagonals.

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