Question 186
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 center does not depend on the position of the points A and D on the respective circles, subject to the assumptions above.
a circle whose center does not depend on the position of the points A and D on the respective circles, subject to the assumptions above.