Question 150
As shown below, two circles $\Gamma_1$, $\Gamma_2$ intersect at points $A$, $B$, one line passing through $B$ intersects $\Gamma_1$, $\Gamma_2$ at points $C$, $D$, another line passing through $B$ intersects $\Gamma_1$, $\Gamma_2$ at points $E$, $F$, and line $CF$ intersects $\Gamma_1$, $\Gamma_2$ at points $P$, $Q$, respectively. Let $M$, $N$ be the middle points of arc $PB$ and arc $QB$, respectively. Prove that if $CD = EF$, then $C$, $F$, $M$, $N$ are concyclic. (Posed by Xiong Bin)
