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{{for|the theorem about the moment of a force|Varignon's theorem (mechanics)}}
[[Image:Varignon parallelogram convex.svg|thumb|300px|Area(''EFGH'') = (1/2)Area(''ABCD'')]]
In [[Euclidean geometry]], '''Varignon's theorem'''
==Theorem==
The midpoints of the sides of an arbitrary quadrilateral form a parallelogram. If the quadrilateral is [[convex polygon|convex]] or [[concave polygon|concave]] (not [[Quadrilateral#Complex_quadrilaterals|complex]]), 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 this area equality also holds for complex quadrilaterals.<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. 52–54, 1967.</ref>
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==Proof==
Referring to the diagram above,
Varignon's theorem can also be 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 (mathematics)|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''
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.
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[[Image:Varignon parallelogram crossed.svg|300px]]
|}
[[File:varignon_parallelogram.svg|thumb|[[Proof without words]] of Varignon's theorem: {{olist
}}]]
==The Varignon parallelogram==
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:<math>m=\tfrac{1}{2}\sqrt{-a^2+b^2-c^2+d^2+p^2+q^2}</math>
where ''p'' and ''q'' are the length of the diagonals.<ref>
:<math>n=\tfrac{1}{2}\sqrt{a^2-b^2+c^2-d^2+p^2+q^2}.</math>
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The lengths of the bimedians can also be expressed in terms of two opposite sides and the distance ''x'' between the midpoints of the diagonals. This is possible when using Euler's quadrilateral theorem in the above formulas. Whence<ref>{{citation
| last = Josefsson
| first = Martin | journal = Forum Geometricorum
| pages = 155–164
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| url = http://forumgeom.fau.edu/FG2011volume11/FG201116.pdf
| volume = 11
| year = 2011
| access-date = 2016-04-05
| archive-date = 2020-01-05
| archive-url = https://web.archive.org/web/20200105031952/http://forumgeom.fau.edu/FG2011volume11/FG201116.pdf
| url-status = dead
}}.</ref>
:<math>m=\tfrac{1}{2}\sqrt{2(b^2+d^2)-4x^2}</math>
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:<math>n=\tfrac{1}{2}\sqrt{2(a^2+c^2)-4x^2}.</math>
In a convex quadrilateral, there is the following [[Duality (mathematics)|dual]] connection between the bimedians and the diagonals:<ref name=Josefsson>{{citation
| last = Josefsson
| first = Martin | journal = Forum Geometricorum
| pages = 13–25
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| url = http://forumgeom.fau.edu/FG2012volume12/FG201202.pdf
| volume = 12
| year = 2012
| access-date = 2012-12-28
| archive-date = 2020-12-05
| archive-url = https://web.archive.org/web/20201205213638/http://forumgeom.fau.edu/FG2012volume12/FG201202.pdf
| url-status = dead
}}.</ref>
* The two bimedians have equal length [[if and only if]] the two diagonals are [[perpendicular]].
* The two bimedians are perpendicular if and only if the two diagonals have equal length.
===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 name=deV>{{citation
| last = de Villiers | first = Michael
| isbn = 9780557102952
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| year = 2009}}.</ref>
The Varignon parallelogram is a [[rectangle]] if and only if the diagonals of the quadrilateral are [[perpendicular]], that is, if the quadrilateral is an [[orthodiagonal quadrilateral]].<ref name=Josefsson/>{{rp|p. 14}} <ref name=deV />{{rp|p. 169}}
For a [[list of self-intersecting polygons|self-crossing]] quadrilateral, the Varignon parallelogram can degenerate to four collinear points, forming a line segment traversed twice. This happens whenever the polygon is formed by replacing two parallel sides of a [[trapezoid]] by the two diagonals of the trapezoid, such as in the [[antiparallelogram]].<ref>{{citation
| last = Muirhead | first = R. F. | author-link = Robert Franklin Muirhead
| date = February 1901
| doi = 10.1017/s0013091500032892
| journal = Proceedings of the Edinburgh Mathematical Society
| pages = 70–72
| title = Geometry of the isosceles trapezium and the contra-parallelogram, with applications to the geometry of the ellipse
| volume = 20| doi-access = free
}}</ref>
==See also==
*[[Perpendicular bisector construction of a quadrilateral]], a different way of forming another quadrilateral from a given quadrilateral
*[[Morley's trisector theorem]], a related theorem on triangles
==Notes==
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== References and further reading ==
*H. S. M. Coxeter, S. L. Greitzer: ''Geometry Revisited''. MAA, Washington 1967, pp. 52-54
*Peter N. Oliver: [
==External links==
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*{{MathWorld|urlname=VarignonsTheorem|title= Varignon's theorem}}
*[http://www.vias.org/comp_geometry/geom_quad_varignon.html Varignon Parallelogram in Compendium Geometry]
* [http://
*[http://www.cut-the-knot.org/Curriculum/Geometry/Varignon.shtml Varignon parallelogram] at cut-the-knot-org
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[[Category:Euclidean geometry]]
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