1999 APMO Problems/Problem 3

Let $\Gamma_1$ and $\Gamma_2$ be two circles intersecting at $P$ and $Q$ . The common tangent, closer to $P$ , of $\Gamma_1$ and $\Gamma_2$ touches $\Gamma_1$ at $A$ and $\Gamma_2$ at $B$ . The tangent of $\Gamma_1$ at $P$ meets $\Gamma_2$ at $C$ , which is different from $P$ , and the extension of $AP$ meets $BC$ at $R$ . Prove that the circumcircle of triangle $PQR$ is tangent to $BP$ and $BR$ .