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]], that 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.

==Theorem==

The midpoints of the sides of an arbitrary quadranglequadrilateral form a parallelogram. If the quadranglequadrilateral is [[convex polygon|convex]] or [[concave|reentrant]], i.e.(the quadrilateral is not a [[Quadrilateral#Self-intersecting quadrilaterals|crossing quadrangle,]]) then the area of the parallelogram is half the area of the quadranglequadrilateral.

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>[[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 or not the quadrilateral is planar or not. ItThe theorem can be generalized to the [[midpoint polygon]] of an arbitrary polygon.

==The Varignon parallelogram==

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

A planar Varignon parallelogram also has the following properties:

*Each pair of opposite sides of the Varignon parallelogram are parallel to a diagonal in the original quadrilateral.

*A side of the Varignon parallelogram is half as long as the diagonal in the original quadrilateral it is parallel to.

*The area of the Varignon parallelogram equals half the area of the original quadrilateral. This is true in convex, concave and crossed quadrilaterals provided the area of the latter is defined to be the difference of the areas of the two triangles it is composed of.<ref>H. S. M. Coxeter and S. L. Greitzer, Geometry Revisited, MAA, 1967, pp. 52-53.</ref>

*The [[perimeter]] of the Varignon parallelogram equals the sum of the diagonals of the original quadrilateral.

*The diagonals of the Varignon parallelogram are the bimedians of the original quadrilateral.

*The two bimedians in a quadrilateral and the line segment joining the midpoints of the diagonals in that quadrilateral are [[Concurrent lines|concurrent]] and are all bisected by their point of intersection.<ref name=Altshiller-Court>Altshiller-Court, Nathan, ''College Geometry'', Dover Publ., 2007.</ref>{{rp|p.125}}

The diagonals of the Varignon parallelogram are the bimedians of the original quadrilateral.

The two bimedians in a quadrilateral and the line segment joining the midpoints of the diagonals in that quadrilateral are [[Concurrent lines|concurrent]] and are all bisected by their point of intersection.<ref name=Altshiller-Court>Altshiller-Court, Nathan, ''College Geometry'', Dover Publ., 2007.</ref>{{rp|p.125}}

In a convex quadrilateral with sides ''a'', ''b'', ''c'' and ''d'', the length of the bimedian that connects the midpoints of the sides ''a'' and ''c'' is