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Geometry · Parallelograms · Similar triangles · Convex quadrilaterals · Midpoints · Parallel lines

Problem 3, 2009

NationalProof

Let \(ABCD\) be a convex quadrilateral and let \(E\) and \(F\) be points on the sides \(AB\) and \(AD\) respectively, chosen so that \(EF \parallel BD\). The segment \(CE\) meets the diagonal \(BD\) at \(G\), and the segment \(CF\) meets the diagonal \(BD\) at \(H\).

Prove that if \(AGCH\) is a parallelogram, then \(ABCD\) is a parallelogram as well.

A B C D E F G H
What the statement builds: \(E\) on \(AB\) and \(F\) on \(AD\) with \(EF \parallel BD\), and the points \(G\), \(H\) cut out of \(BD\) by \(CE\) and \(CF\). The hypothesis that \(AGCH\) is a parallelogram is an extra assumption and is not drawn here.

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Slovenian High School Mathematics Competition for Vega Awards (MaSSA), drzavno (national) round 2009, 1. letnik, category A, problem 3. Organized by DMFA Slovenije (Society of Mathematicians, Physicists and Astronomers of Slovenia). Source

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