US-TST 2001 Problem 7

Let $ABCD$ be a convex quadrilateral such that $\angle ABC = \angle ADC = 135^{\circ}$ and $$AC^2\cdot BD^2 = 2\cdot AB\cdot BC\cdot CD\cdot DA.$$ Prove that the diagonals of the quadrilateral $ABCD$ are perpendicular.