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Line ZN is parallel to the bases of the trapezoid

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

July 20, 2008
geometrytrapezoidanalytic geometryratioperpendicular bisectorangle bisectorprojective geometry

Problem Statement

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