Question 55
Given two circles $(O), (O')$ with different radii and intersecting at $A, B$. Circle of center $A$, radius $AB$ intersects $(O), (O')$ again at $C, D$ respectively. Let $EF$ be the common tangent closer to $B$ of the two circles with $E \in (O), F \in (O')$. Rays $AE, AF$ intersect $BC, BD$ at $M, N$ respectively. Prove that the internal bisector of $\angle CBD$ passes through the circumcenter of triangle $AMN$.