This result is a possible statement of the dual of Pappus's hexagon theorem (the statement on the Wikipedia is different, but the situation is the same: they give names to lines, while you give names to points).

With this information, you should be able to find a proof on the net. There's for instance an idea for a direct proof (not a standard one) in a reference given in the wikipedia article ; otherwise, I think it is more usual to prove Pappus's theorem (you'll find plenty of proofs for that) and obtain your result by duality (if you know projective geometry). If this is an homework, don't you have any given hints?