DANUBE 2008 Problem 2

In a triangle $ABC$ let $A_1$ be the midpoint of side $BC$ . Draw circles with centers $A, A1$ and radii $AA_1, BC$ respectively and let $A'A''$ be their common chord. Similarly denote the segments $B'B''$ and $C'C''$ . Show that lines $A'A'', B'B'''$ and $C'C''$ are concurrent.