FKMO 2007 Problem 1

Let $O$ be the circumcenter of an acute triangle $ABC$ and let $k$ be the circle with center $P$ that is tangent to $O$ at $A$ and tangent to side $BC$ at $D$ . Circle $k$ meets $AB$ and $AC$ again at $E$ and $F$ respectively. The lines $OP$ and $EP$ meet $k$ again at $I$ and $G$ . Lines $BO$ and $IG$ intersect at $H$ . Prove that $\frac{{DF}^2}{AF}=GH$ .