Suppose that are the consecutive vertices of the quadrilateral.
Let midpoints be .
We know that .
Do the same thing for
You will be done.
The midpoints of the adjacent sides of a quadrilateral are joined. Show that the resulting figure is a parallelogram.
I know that you suppose to somehow use vectors to prove this, but I dont know how you are suppose to model this?
I have given you the standard vector proof.
BTW. This theorem is true for any quadrilateral, even non-convex quadrilaterals.
And the proof is the same.
I suspect that you do not understand vector addition.
If that is the case, you no chance proving this.
As I said, the proof I gave is the standard.