Orthogonality in Two Squares

21 October 2015, Created with GeoGebra

$AMNP$ and $ABCD$ are two squares.

Orthogonality in Two Squares - problem

Prove that $BM\perp DP.$

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

Orthogonality in Two Squares - solution

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

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