Editing Parallelogram (section)

==Parallelograms arising from other figures==

===Automedian triangle===
An [[automedian triangle]] is one whose [[median (geometry)|medians]] are in the same proportions as its sides (though in a different order). If ''ABC'' is an automedian triangle in which vertex ''A'' stands opposite the side ''a'',  ''G'' is the [[centroid]] (where the three medians of ''ABC'' intersect), and ''AL'' is one of the extended medians of ''ABC'' with ''L'' lying on the circumcircle of ''ABC'', then ''BGCL'' is a parallelogram.

===Varignon parallelogram===
{{main|Varignon's theorem}}

[[File:varignon_parallelogram.svg|thumb|Proof without words of Varignon's theorem ]]

[[Varignon's theorem]] holds that the [[midpoint]]s of the sides of an arbitrary quadrilateral are the vertices of a parallelogram, called its ''Varignon parallelogram''. If the quadrilateral is [[Convex polygon|convex]] or [[Concave polygon|concave]] (that is, not self-intersecting), then the area of the Varignon parallelogram is half the area of the quadrilateral.

[[Proof without words]] (see figure):

# An arbitrary quadrilateral and its diagonals.
# Bases of similar triangles are parallel to the blue diagonal.
# Ditto for the red diagonal.
# The base pairs form a parallelogram with half the area of the quadrilateral, ''A<sub>q</sub>'', as the sum of the areas of the four large triangles, ''A<sub>l</sub>'' is 2 ''A<sub>q</sub>'' (each of the two pairs reconstructs the quadrilateral) while that of the small triangles, ''A<sub>s</sub>'' is a quarter of ''A<sub>l</sub>'' (half linear dimensions yields quarter area), and the area of the parallelogram is ''A<sub>q</sub>'' minus ''A<sub>s</sub>''.

===Tangent parallelogram of an ellipse===

For an [[ellipse]], two diameters are said to be [[Conjugate diameters|conjugate]] if and only if the [[tangent line]] to the ellipse at an endpoint of one diameter is parallel to the other diameter. Each pair of conjugate diameters of an ellipse has a corresponding [[tangent parallelogram]], sometimes called a bounding parallelogram, formed by the tangent lines to the ellipse at the four endpoints of the conjugate diameters. All tangent parallelograms for a given ellipse have the same area.

It is possible to [[Compass and straightedge constructions|reconstruct]] an ellipse from any pair of conjugate diameters, or from any tangent parallelogram.

===Faces of a parallelepiped===

A [[parallelepiped]] is a three-dimensional figure whose six [[face (geometry)|faces]] are parallelograms.