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s[ABP]+s[CDQ]=s[MNPQ]

Source: Moldova NMO 2002 grade 9 problem nr.4

November 2, 2008

Problem Statement

Let ABCD ABCD be a convex quadrilateral and let N N on side AD AD and M M on side BC BC be points such that \dfrac{AN}{ND}\equal{}\dfrac{BM}{MC}. The lines AM AM and BN BN intersect at P P, while the lines CN CN and DM DM intersect at Q Q. Prove that if S_{ABP}\plus{}S_{CDQ}\equal{}S_{MNPQ}, then either AD∥BC AD\parallel BC or N N is the midpoint of DA DA.
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