Parallelogram from Quadrilaterals

It is known that in any triangle, the segment that joins the midpoints of two adjacent sides is parallel to the third side. For example, for $\triangle ABC$ in our figure, segment $MN$ joins the midpoints of $AB$ and $BC$ , and is parallel to $AC$ .

So from $\triangle ABC$ and $\triangle CDA$ we have:

$MN || AC \wedge PQ || AC \Rightarrow MN || PQ$

And from $\triangle BCD$ and $\triangle DAB$ we have:

$NP || BD \wedge QM || BD \Rightarrow NP || QM$

And because $MN || PQ$ and $NP || QM$ , we conclude that $MNPQ$ is indeed a parallelogram.

Because we proved this by observing triangles formed by the vertexes of our quadrilateral, this applies not only to convex quadrilaterals, but also to any arrangement of four points such that no three of them are colinear: