TUYMAADA 2008 J6

Let $ABCD$ be an isosceles trapezoid with $AD \parallel BC$ . Its diagonals $AC$ and $BD$ intersect at point $M$ . Points $X$ and $Y$ on the segment $AB$ are such that $AX = AM$ , $BY = BM$ . Let $Z$ be the midpoint of $XY$ and $N$ is the point of intersection of the segments $XD$ and $YC$ . Prove that the line $ZN$ is parallel to the bases of the trapezoid.

Author: A. Akopyan, A. Myakishev