RO-NMO 2004 Problem 73

Let $ABCD$ be an orthodiagonal trapezoid such that $\measuredangle A = 90^{\circ}$ and $AB$ is the larger base. The diagonals intersect at $O$ , $\left( OE \right.$ is the bisector of $\measuredangle AOD$ , $E \in \left( AD \right)$ and $EF \| AB$ , $F \in \left( BC \right)$ . Let $P,Q$ the intersections of the segment $EF$ with $AC,BD$ . Prove that:

(a) $EP=QF$ ;

(b) $EF=AD$ .

Claudiu-Stefan Popa