OMCC 2010 Problem 6

Let $\Gamma$ and $\Gamma_1$ be two circles internally tangent at $A$ , with centers $O$ and $O_1$ and radii $r$ and $r_1$ , respectively ( $r>r_1$ ). $B$ is a point diametrically opposed to $A$ in $\Gamma$ , and $C$ is a point on $\Gamma$ such that $BC$ is tangent to $\Gamma_1$ at $P$ . Let $A'$ the midpoint of $BC$ . Given that $O_1A'$ is parallel to $AP$ , find the ratio $r/r_1$ .