ZA-SAMO 2002 Problem 3

A small square $PQRS$ is contained in a big square. Produce $PQ$ to $A$ , $QR$ to $B$ , $RS$ to $C$ and $SP$ to $D$ so that $A$ , $B$ , $C$ and $D$ lie on the four sides of the large square in order, produced if necessary. Prove that $AC = BD$ and $AC \perp BD$ .