Question 46
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 $AA'$. Let $\gamma_B$ be the circle through $B$ and $X$, centered on the line $BC$, and let $\gamma_C$ be the circle through $C$ and $X$, centered on the line $BC$. The circle $\gamma_B$ meets the lines $AB$ and $BB'$ again at $M$ and $M'$, respectively, and the circle $\gamma_C$ meets the lines $AC$ and $CC'$ again at $N$ and $N'$, respectively. Show that the points $M$, $M'$, $N$ and $N'$ are collinear.
