Let $M, N, P, Q$ be points on sides $AB, BC, CD, DA$ of a convex quadrilateral $ABCD$ such that $AQ = DP = CN = BM$. Prove that if $MNPQ$ is a square, then $ABCD$ is also a square.
Source: Romania TST 1995 2.3
Tags: geometry, square
Let $M, N, P, Q$ be points on sides $AB, BC, CD, DA$ of a convex quadrilateral $ABCD$ such that $AQ = DP = CN = BM$. Prove that if $MNPQ$ is a square, then $ABCD$ is also a square.