Question 30
The circles $C_1$ and $C_2$ intersect at two distinct points $A$ and $K$. The common tangent to $C_1$ and $C_2$ closer 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 reflections of $K$ in the lines $PQ$ and $BC$ respectively, prove that the points $A$, $E$ and $F$ are collinear.