I am trying to prove: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram, by using vectors.

This is what I have:

Let ABCD be a quadrilateral with diagonals AC, BD bisecting each other.

Then DC =1/2 (AC - BD),

and AB = 1/2 (DB - CA).

How do I equate these? Am I allowed to say -AB = BA = 1/2 (AC - BD)?

Thanks for your help.