RO-TST 2008 Problem 10

Let $ABC$ be a triangle with $\measuredangle{BAC} < \measuredangle{ACB}$ . Let $D$ , $E$ be points on the sides $AC$ and $AB$ , such that the angles $ACB$ and $BED$ are congruent. If $F$ lies in the interior of the quadrilateral $BCDE$ such that the circumcircle of triangle $BCF$ is tangent to the circumcircle of $DEF$ and the circumcircle of $BEF$ is tangent to the circumcircle of $CDF$ , prove that the points $A$ , $C$ , $E$ , $F$ are concyclic.

Author: Cosmin Pohoata