KöMaL 2005 B3869

Point $M$ lies on the angle bisector drawn from vertex $A$, inside an acute-angled triangle ABC . The other intersections of the lines $AM$, $BM$, $CM$ with the circumscribed circle are $A_1$, $B_1$ and $C_1$, respectively. The lines $AB$ and $C_1 A_1$ intersect at $L$, and the lines $AC$ and $B_1 A_1$ intersect at $N$. Prove that $LN$ is parallel to $BC$.