Varignon's theorem: Difference between revisions

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{{for|the theorem about the momentum of a force|Varignon's theorem (mechanics)}}

[[Image:Varignon parallelogram convex.svg|thumb|300px|Area(''EFGH'') = (1/2)Area(''ABCD''); ''EF'' and ''HG'' are parallel, as are ''EH'' and ''FG'']]

'''Varignon's theorem''' is a statement in [[Euclidean geometry]] by [[Pierre Varignon]], that was first published in 1731. It 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 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 the area of the quadrangle''.

==Theorem==

: ''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 the area of the quadrangle''.

If one introduces the concept of oriented areas for [[Polygon|n-gons]], then the area equality above holds for crossed quadrilaterals as well.<ref>[[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>

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The Varignon parallelogram exists even for a [[Quadrilateral#More_quadrilaterals|skew quadrilateral]], and is planar whether or not the quadrilateral is planar. It can be generalized to the [[midpoint polygon]] of an arbitrary polygon.

==The Varignon parallelogram==

===Properties===

<!-- copied from [[quadrilateral#bimedian]] -->

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* The two bimedians are perpendicular if and only if the two diagonals have equal length.

===Special cases===

==Special cases==

The Varignon parallelogram is a [[rhombus]] if and only if the two diagonals of the quadrilateral have equal length, that is, if the quadrilateral is an [[equidiagonal quadrilateral]].<ref>{{citation

| last = de Villiers | first = Michael