2005 CMO 第 1 题

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.

圆与三角形 $ABC$ 的三条边 $BC,CA,AB$ 相交于点 $D_1,D_2;E_1,E_2;分别为F_1,F_2$。此外，线段$D_1E_1$和$D_2F_2$相交于点$L$，线段$E_1F_1$和$E_2D_2$相交于点$M$，线段$F_1D_1$和$F_2E_2$相交于点$N$。证明 $AL,BM,CN$ 行是并发的。