HMIC 2013 Problem 3

Triangle $A B C$ is inscribed in a circle $\omega$ such that $\angle A=60^{\circ}$ and $\angle B=75^{\circ}$. Let the bisector of angle $A$ meet $B C$ and $\omega$ at $E$ and $D$, respectively. Let the reflections of $A$ across $D$ and $C$ be $D^{\prime}$ and $C^{\prime}$, respectively. If the tangent to $\omega$ at $A$ meets line $B C$ at $P$, and the circumcircle of $A P D^{\prime}$ meets line $A C$ at $F \neq A$, prove that the circumcircle of $C^{\prime} F E$ is tangent to $B C$ at $E$.