Prove, in this figure, that $EFGH$ is a parallelogram

It is well-known (and easily provable) that the "mid-point polygon" of any quadrilateral is a parallelogram. (Opposite sides are parallel to a diagonal.) The Question's $EFGH$ is simply the dilation of the mid-point polygon with respect to point $O$, with scale factor $2$, and is therefore also a parallelogram.