SMO 2004 Problem 3

Let $AD$ be the common chord of two circles $\Gamma_1$ and $\Gamma_2$ . A line through $D$ intersects $\Gamma_1$ at $B$ and $\Gamma_2$ at $C$ . Let $E$ be a point on the segment $AD$ , different from $A$ and $D$ . The line $CE$ intersect $\Gamma_1$ at $P$ and $Q$ . The line $BE$ intersects $\Gamma_2$ at $M$ and $N$ . (i) Prove that $P,Q,M,N$ lie on the circumference of a circle $\Gamma_3$ . (ii) If the centre of $\Gamma_3$ is $O$ , prove that $OD$ is perpendicular to $BC$ .