OMM 2014 Problem 3

Let $\Gamma_1$ be a circle and $P$ a point outside of $\Gamma_1$ . The tangents from $P$ to $\Gamma_1$ touch the circle at $A$ and $B$ . Let $M$ be the midpoint of $PA$ and $\Gamma_2$ the circle through $P$ , $A$ and $B$ . Line $BM$ cuts $\Gamma_2$ at $C$ , line $CA$ cuts $\Gamma_1$ at $D$ , segment $DB$ cuts $\Gamma_2$ at $E$ and line $PE$ cuts $\Gamma_1$ at $F$ , with $E$ in segment $PF$ . Prove lines $AF$ , $BP$ , and $CE$ are concurrent.