# Orthogonality in Two Squares

### Problem

$AMNP$ and $ABCD$ are two squares.

Prove that $BM\perp DP.$

### Solution

Observe that the counterclockwise (in the diagram below) rotation through $90^{\circ}$ around point $A$ brings $P$ to $M$ and $D$ to $B.$

Thus, the entire $\Delta ADP$ maps onto $\Delta ABM,$ implying, in particular, that their third sides are also perpendicular: $BM\perp DP.$