NL-NWO 1998 Problem 4

Let $ABCD$ be a convex quadrilateral such that $AC \perp BD$ .

(a) Prove that $AB^2 + CD^2 = BC^2 + DA^2$ .

(b) Let $PQRS$ be a convex quadrilateral such that $PQ = AB$ , $QR = BC$ , $RS = CD$ and $SP = DA$ . Prove that $PR \perp QS$ .