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ratio of triangle area indepedent of choices of 2 points, ratios in cyclic

Source: Polish second round 1999 p3

January 19, 2020
ratiogeometrycyclic quadrilateralarea of a triangleareas

Problem Statement

Let ABCDABCD be a cyclic quadrilateral and let EE and FF be the points on the sides ABAB and CDCD respectively such that AE:EB=CF:FDAE : EB = CF : FD. Point PP on the segment EF satsfies EP:PF=AB:CDEP : PF = AB : CD. Prove that the ratio of the areas of △APD\vartriangle APD and △BPC\vartriangle BPC does not depend on the choice of EE and FF.
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