JBMO-SL 2006 Problem 11

Circles $\mathcal{C}_1$ and $\mathcal{C}_2$ intersect at $A$ and $B$ . Let $M\in AB$ . A line through $M$ (different from $AB$ ) cuts circles $\mathcal{C}_1$ and $\mathcal{C}_2$ at $Z,D,E,C$ respectively such that $D,E\in ZC$ . Perpendiculars at $B$ to the lines $EB,ZB$ and $AD$ respectively cut circle $\mathcal{C}_2$ in $F,K$ and $N$ . Prove that $KF=NC$ .