Problem

Source: USA TST (42nd IMO-2001)

Tags: geometry, rhombus, inequalities, geometry solved



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.