RO-NMO 2006 Problem 92

Let $ABC$ and $DBC$ be isosceles triangle with the base $BC$ . We know that $\measuredangle ABD = \frac{\pi}{2}$ . Let $M$ be the midpoint of $BC$ . The points $E,F,P$ are chosen such that $E \in (AB)$ , $P \in (MC)$ , $C \in (AF)$ , and $\measuredangle BDE = \measuredangle ADP = \measuredangle CDF$ . Prove that $P$ is the midpoint of $EF$ and $DP \perp EF$ .