Question 47
Two circles $\Gamma,\Gamma'$ meet at distinct points $A,B$. A line through $B$ meets $\Gamma,\Gamma'$ again at $C,D$, respectively, with $B$ between $C,D$. Another line through $B$ meets them again at $E,F$, respectively, with $E$ between $B,F$. Suppose $|CD|=|EF|$. The interior of segment $CF$ meets $\Gamma,\Gamma'$ at $P,Q$, respectively. Let $M,N$ be the midpoints of the arcs $PB$ not containing $C$ on $\Gamma$ and $BQ$ not containing $F$ on $\Gamma'$, respectively. Prove that $CNMF$ is cyclic.