KöMaL 2009 B4165

The diagonals of a convex quadrilateral ABCD intersect at $O$. Prove that $AB^2$+$BC^2$+$CD^2$+$DA^2$=2($AO^2$+$BO^2$+$CO^2$+$DO^2$) is true if and only if the diagonals $AC$ and $BD$ are perpendicular or one of them has its midpoint at $O$.