BALKAN-SL 2010 G4

Let $ABC$ be a given triangle and $\ell$ be a line that meets the lines $BC, CA$ and $AB$ in $A_1,B_1$ and $C_1$ respectively. Let $A'$ be the midpoint, of the segment connecting the projections of $A_1$ onto the lines $AB$ and $AC$ . Construct, analogously the points $B'$ and $C'$ . (a) Show that the points $A', B'$ and $C'$ are collinear on some line $\ell'$ . (b) Show that if $\ell$ contains the circumcenter of the triangle $ABC$ , then $\ell'$ contains the center of it's Euler circle.