National problem · Geometry

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.

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.