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← Drzavno tekmovanje (national round) 2002National

Geometry · Trapezoid · Homothety · Parallel lines · Ratios of segments · Midpoints

Problem 3, 2002

NationalProof

Let \(M\) be the midpoint of the base \(AB\) of a trapezoid \(ABCD\), so that \(AB \parallel CD\). A point \(E\) is taken in the interior of the segment \(AC\) in such a way that the lines \(BC\) and \(ME\) meet at a point \(F\), the lines \(FD\) and \(AB\) meet at a point \(G\), and the lines \(DE\) and \(AB\) meet at a point \(H\). Prove that \(M\) is the midpoint of the segment \(GH\).

G A M B H D C E F
The configuration. The statement fixes no position for \(E\) inside \(AC\), so the picture is only one of several possible shapes.

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Slovenian High School Mathematics Competition for Vega Awards (MaSSA), drzavno (national) round 2002, 1. letnik, category A, problem 3. Organized by DMFA Slovenije (Society of Mathematicians, Physicists and Astronomers of Slovenia). Source

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