USAMO 2012 Problem 5

Let $P$ be a point in the plane of $\triangle ABC$ , and $\gamma$ a line passing through $P$ . Let $A', B', C'$ be the points where the reflections of lines $PA, PB, PC$ with respect to $\gamma$ intersect lines $BC, AC, AB$ respectively. Prove that $A', B', C'$ are collinear.