OMCC 2006 Problem 6

Let $ABCD$ be a convex quadrilateral. $I=AC\cap BD$ , and $E$ , $H$ , $F$ and $G$ are points on $AB$ , $BC$ , $CD$ and $DA$ respectively, such that $EF \cap GH= I$ . If $M=EG \cap AC$ , $N=HF \cap AC$ , show that $$\frac{AM}{IM}\cdot \frac{IN}{CN}=\frac{IA}{IC}.$$