Let O be the intersection of the diagonals.
It's easy to prove that the triangles OAF and OBC are similar.
And so are OAD and OBE.
Use these facts to prove that OFE and ODC are similar
Otherwise, you can use vectors.
In a quad. ABCD, a line AF parallel to BC meets BD in F and a line BE parallel to AD meets AC in E. Prove that EF is parallel to CD.
I think this is how it looks like:
Could someone please give me some hints on how to start this problem?