Varignon's theorem

Varignon's theorem is a statement in Euclidean geometry, that deals with the construction of a particular parallelogram, the Varignon parallelogram, from an arbitrary quadrilateral (quadrangle). It is named after Pierre Varignon, who published it in 1731.

The midpoints of the sides of an arbitrary quadrilateral form a parallelogram. If the quadrilateral is convex or reentrant, (the quadrilateral is not a crossing quadrangle) then the area of the parallelogram is half the area of the quadrilateral.

If one introduces the concept of oriented areas for n-gons, then the area equality above also holds for crossed quadrilaterals.

The Varignon parallelogram exists even for a 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.

Varignon's theorem is easily proved as a theorem of affine geometry organized as linear algebra with the linear combinations restricted to coefficients summing to 1, also called affine or barycentric coordinates. The proof applies even to skew quadrilaterals in spaces of any dimension.

Any three points E, F, G are completed to a parallelogram (lying in the plane containing E, F, and G) by taking its fourth vertex to be E − F + G. In the construction of the Varignon parallelogram this is the point (A + B)/2 − (B + C)/2 + (C + D)/2 = (A + D)/2. But this is the point H in the figure, whence EFGH forms a parallelogram.

In short, the centroid of the four points A, B, C, D is the midpoint of each of the two diagonals EG and FH of EFGH, showing that the midpoints coincide.

A second proof requires less algebra. By drawing in the diagonals of the quadrilateral, we notice two triangles are created for each diagonal. And by the midpoint theorem, the segment containing two midpoints of adjacent sides is both parallel and half the respective diagonal. Since two opposite sides are equal and parallel, we have that the quadrilateral must be a parallelogram.

...
Wikipedia