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Product of ratios

Source: Central American Olympiad 2006, Problem 6

April 30, 2007
ratiotrigonometryprojective geometrygeometry proposedgeometry

Problem Statement

Let ABCDABCD be a convex quadrilateral. I=AC∩BDI=AC\cap BD, and EE, HH, FF and GG are points on ABAB, BCBC, CDCD and DADA respectively, such that EF∩GH=IEF \cap GH= I. If M=EG∩ACM=EG \cap AC, N=HF∩ACN=HF \cap AC, show that AMIM⋅INCN=IAIC.\frac{AM}{IM}\cdot \frac{IN}{CN}=\frac{IA}{IC}.
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