US-TST 2006 Problem 2

In acute triangle $ABC$ , segments $AD; BE$ , and $CF$ are its altitudes, and $H$ is its orthocenter. Circle $\omega$ , centered at $O$ , passes through $A$ and $H$ and intersects sides $AB$ and $AC$ again at $Q$ and $P$ (other than $A$ ), respectively. The circumcircle of triangle $OPQ$ is tangent to segment $BC$ at $R$ . Prove that $\frac{CR}{BR}=\frac{ED}{FD}.$