RO-TST 2010 Problem 4

Two circles in the plane, $\gamma_1$ and $\gamma_2$ , meet at points $M$ and $N$ . Let $A$ be a point on $\gamma_1$ , and let $D$ be a point on $\gamma_2$ . The lines $AM$ and $AN$ meet again $\gamma_2$ at points $B$ and $C$ , respectively, and the lines $DM$ and $DN$ meet again $\gamma_1$ at points $E$ and $F$ , respectively. Assume the order $M$ , $N$ , $F$ , $A$ , $E$ is circular around $\gamma_1$ , and the segments $AB$ and $DE$ are congruent. Prove that the points $A$ , $F$ , $C$ and $D$ lie on a circle whose centre does not depend on the position of the points $A$ and $D$ on the respective circles, subject to the assumptions above.