Question

prove using vectors of midpoints of two opposite sides of a quadrilateral and the mid points of the diagonal are the vertices of a parallelogram ​

Answer 1

Answer:

As Jordan pointed out in his answer above, if the non-bisected side of the quadrilateral happen to be parallel, you have a degenerate case where the parallelogram described is “flattened” so a sides are on top of each other.

With that caveat,in classical geometry, it is easy to prove you have a parallelogram with the Midsegment Theorem. I think the classical approach would be easier to explain. [Show MP and QN both 1/2 AD and MQ and PN both 1/2 BC.] Attached is my vector proof but remember I don’t like vector proofs.