Geometry · Trapezoid · Ratios of areas · Intercept theorem · Midpoints
Let \(ABCD\) be a trapezoid with \(AB \parallel CD\), and let \(P\) be a point on the extension of the diagonal \(AC\) beyond \(C\), so that \(C\) lies between \(A\) and \(P\). Let \(X\) and \(Y\) be the midpoints of the parallel sides \(AB\) and \(CD\), and let \(M\) and \(N\) be the points in which the lines \(PX\) and \(PY\) meet the sides \(BC\) and \(DA\), respectively. Prove that the line \(MN\) is parallel to the parallel sides of the trapezoid.
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Serbian Republic Competition (Republicko takmicenje) 2004, high school grade I, category A, problem 1. Organized by the Mathematical Society of Serbia (DMS). Source