Problem

Source: 2002 Romania JBMO TST1 p4

Tags: geometry, parallelogram, square, similar triangles, similar



Let $ABCD$ be a parallelogram of center $O$. Points $M$ and $N$ are the midpoints of $BO$ and $CD$, respectively. Prove that if the triangles $ABC$ and $AMN$ are similar, then $ABCD$ is a square.