OIM 2016 Problem 5

The circumferences $C_1$ and $C_2$ cut each other at different points $A$ and $K$ . The common tangent to $C_1$ and $C_2$ nearer to $K$ touches $C_1$ at $B$ and $C_2$ at $C$ . Let $P$ be the foot of the perpendicular from $B$ to $AC$ , and let $Q$ be the foot of the perpendicular from $C$ to $AB$ . If $E$ and $F$ are the symmetric points of $K$ with respect to the lines $PQ$ and $BC$ , respectively, prove that $A, E$ and $F$ are collinear.