A Varignon parallelogram is the figure formed when the midpoints of adjacent sides of a quadrilateral are joined in boundary order. Varignon's theorem demonstrated that this figure is a parallelogram. The center of the Varignon parallelogram is the geometric centroid of four point masses placed on the vertices of the quadrilateral.
For a convex quadrilateral, each side of its Varignon parallelogram is parallel to one of the original diagonals and is half as long. The Varignon parallelogram is a rhombus iff the original quadrilateral is an equidiagonal quadrilateral. It is a rectangle iff the original quadrilateral is an orthodiagonal quadrangle, and a square iff the original quadrilateral is a midsquare quadrilateral (Josefsson 2014, pp. 130-131 and 137).
The center of the Varignon parallelogram lies on a polygon diagonal of the original convex quadrilateral iff the latter is a bisect-diagonal quadrilateral (Josefsson 2017, p. 216).
