OMCC 2014 Problem 2

Let $ABCD$ be a trapezoid with bases $AB$ and $CD$ , inscribed in a circle of center $O$ . Let $P$ be the intersection of the lines $BC$ and $AD$ . A circle through $O$ and $P$ intersects the segments $BC$ and $AD$ at interior points $F$ and $G$ , respectively. Show that $BF=DG$ .