Problem

Source: Tuymaada 2008, Junior League, Second Day, Problem 6.

Tags: geometry, trapezoid, analytic geometry, ratio, perpendicular bisector, angle bisector, projective geometry



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