National problem · Geometry

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 \(GH\).

The configuration. The statement fixes no position for \(E\) inside \(AC\), so the picture is only one of several possible shapes.