CGMO 2009 Problem 2

Right triangle $ABC,$ with $\angle A=90^{\circ},$ is inscribed in circle $\Gamma.$ Point $E$ lies on the interior of arc ${BC}$ (not containing $A$ ) with $EA>EC.$ Point $F$ lies on ray $EC$ with $\angle EAC = \angle CAF.$ Segment $BF$ meets $\Gamma$ again at $D$ (other than $B$ ). Let $O$ denote the circumcenter of triangle $DEF.$ Prove that $A,C,O$ are collinear.