User:Kmhkmh/sandbox

From Wikipedia, the free encyclopedia

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 there 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.

Useful stubs:

Limay River
A sky view of the Limay River
A sky view of the Limay River
Origin Nahuel Huapi Lake
Mouth Río Negro
Basin countries Argentina
Length 380 km
Source elevation 770 m AMSL
Avg. discharge 700 m³/s
Basin area 61,723 km²

Eric W. Weisstein, Morrie's Law at MathWorld.