The midpoints of the sides of an arbitrary quadrilateral form a parallelogram. If the quadrilateral is [[convex polygon|convex]] or [[concave polygon|reentrantconcave]], (the quadrilateral is not a [[Quadrilateral#Self-intersecting quadrilateralsComplex_quadrilaterals|crossing quadranglecomplex]]), then the area of the parallelogram is half the area of the quadrilateral.
If one introduces the concept of oriented areas for [[Polygon|''n''-gons]], then thethis area equality above also holds for crossedcomplex quadrilaterals.<ref name=Coxeter>[[Coxeter|Coxeter, H. S. M.]] and Greitzer, S. L. "Quadrangle; Varignon's theorem" §3.1 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 52–54, 1967.</ref>
The Varignon parallelogram exists even for a [[Quadrilateral#More_quadrilateralsSkew_quadrilaterals|skew quadrilateral]], and is planar whether the quadrilateral is planar or not. The theorem can be generalized to the [[midpoint polygon]] of an arbitrary polygon.
