Varignon's theorem: Difference between revisions

Revision as of 16:39, 16 October 2016

Revision as of 15:35, 25 October 2016

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The midpoints of the sides of an arbitrary quadrilateral form a parallelogram. If the quadrilateral is [[convex polygon|convex]] or [[concave polygon|reentrant]], (the quadrilateral is not a [[Quadrilateral#Self-intersecting quadrilaterals|crossing quadrangle]]) 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 the area equality above also holds for crossed quadrilaterals as well.<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&ndash;54, 1967.</ref>

The Varignon parallelogram exists even for a [[Quadrilateral#More_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.