CMO 2005 Problem 2

A circle meets the three sides $BC,CA,AB$ of a triangle $ABC$ at points $D_1,D_2;E_1,E_2; F_1,F_2$ respectively. Furthermore, line segments $D_1E_1$ and $D_2F_2$ intersect at point $L$ , line segments $E_1F_1$ and $E_2D_2$ intersect at point $M$ , line segments $F_1D_1$ and $F_2E_2$ intersect at point $N$ . Prove that the lines $AL,BM,CN$ are concurrent.