OSN 2015 Problem 6

Let $ABC$ be an acute angled triangle with circumcircle $O$ . Line $AO$ intersects the circumcircle of triangle $ABC$ again at point $D$ . Let $P$ be a point on the side $BC$ . Line passing through $P$ perpendicular to $AP$ intersects lines $DB$ and $DC$ at $E$ and $F$ respectively . Line passing through $D$ perpendicular to $BC$ intersects $EF$ at point $Q$ . Prove that $EQ = FQ$ if and only if $BP = CP$ .