2016 OIM Problems/Problem 5

Problem

The circles $\Gamma_1$ and $\Gamma_2$ intersect at two different points $A$ and $K$ . The tangent common to $\Gamma_1$ and $\Gamma_2$ closest to 4K $touches$ \Gamma_1 $at$ B $and$ \Gamma_2 $at$ C $. Let$ P $be the foot of the perpendicular from$ B $on$ AC $, and$ Q $the foot of the perpendicular from$ C $on$ AB $. If$ E $and$ F $are the symmetrical points of$ K $with respect to the lines$ PQ $and$ BC $, prove that the points$ A, E $and$ F$ are collinear.