OMM 2004 Problem 5

Let $\omega_1$ and $\omega_2$ be two circles such that the center $O$ of $\omega_2$ lies in $\omega_1$ . Let $C$ and $D$ be the two intersection points of the circles. Let $A$ be a point on $\omega_1$ and let $B$ be a point on $\omega_2$ such that $AC$ is tangent to $\omega_2$ in C and BC is tangent to $\omega_1$ in $C$ . The line segment $AB$ meets $\omega_2$ again in $E$ and also meets $\omega_1$ again in F. The line $CE$ meets $\omega_1$ again in $G$ and the line $CF$ meets the line $GD$ in $H$ . Prove that the intersection point of $GO$ and $EH$ is the center of the circumcircle of the triangle $DEF$ .