RO-NMO 2005 Problem 73

Let $ABCD$ be a quadrilateral with $AB\parallel CD$ and $AC \perp BD$ . Let $O$ be the intersection of $AC$ and $BD$ . On the rays $(OA$ and $(OB$ we consider the points $M$ and $N$ respectively such that $\angle ANC = \angle BMD = 90^\circ$ . We denote with $E$ the midpoint of the segment $MN$ . Prove that

a) $\triangle OMN \sim \triangle OBA$ ; b) $OE \perp AB$ .

Claudiu-Stefan Popa