Varignon's theorem

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A(\Box EFGH)=\frac{1}{2}A(\Box ABCD)

Varignon's theorem is a statement in Euclidean geometry that was first published by Pierre Varignon in 1731. It deals with the construction of particular parallelogram (Varignon parallelogram) from an arbitrary quadrangle.

The midpoints of the sides of an arbitrary quadrangle form a parallelogram. If the quadrangle is convex or reentrant, i.e. not a crossing quadrangle, then the area of the parallelogram is half as big as the area of the quadrangle.

If one introduces the concept or oriented areas for n-gons, then the area of a crossed quadrangle can be defined in such a way, that the area equality above holds for crossed quadrangles as well.[1]


convex quadrangle reentrant quadrangle crossing quadrangle

[edit] References

  1. ^ 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.