RO-NMO 2005 Problem 71

Let $ABCD$ be a parallelogram. The interior angle bisector of $\angle ADC$ intersects the line $BC$ in $E$ , and the perpendicular bisector of the side $AD$ intersects the line $DE$ in $M$ . Let $F= AM \cap BC$ . Prove that:

a) $DE=AF$ ; b) $AD\cdot AB = DE\cdot DM$ .

Daniela and Marius Lobaza, Timisoara