RU-ORAL-GEOMETRY 2022 Problem 3

In quadrilateral $ABCD$ , sides $AB$ and $CD$ are equal (but not parallel), points $M$ and $N$ are the midpoints of $AD$ and $BC$ . The perpendicular bisector of $MN$ intersects sides $AB$ and $CD$ at points $P$ and $Q$ , respectively. Prove that $AP = CQ$ .

(M. Kungozhin)