RO-TST 2014 Problem 1

Let $ABC$ be a triangle, let ${A}'$ , ${B}'$ , ${C}'$ be the orthogonal projections of the vertices $A$ , $B$ , $C$ on the lines $BC$ , $CA$ and $AB$ , respectively, and let $X$ be a point on the line $A{A}'$ .Let $\gamma_{B}$ be the circle through $B$ and $X$ , centred on the line $BC$ , and let $\gamma_{C}$ be the circle through $C$ and $X$ , centred on the line $BC$ .The circle $\gamma_{B}$ meets the lines $AB$ and $B{B}'$ again at $M$ and ${M}'$ , respectively, and the circle $\gamma_{C}$ meets the lines $AC$ and $C{C}'$ again at $N$ and ${N}'$ , respectively.Show that the points $M$ , ${M}'$ , $N$ and ${N}'$ are collinear.