Editing Quadrilateral (section)

==Bimedians==
{{See also|Varignon's theorem}}
[[File:Varignon theorem convex.png|300px|thumb|The Varignon
 parallelogram ''EFGH'']]

The [[Quadrilateral#Special line segments|bimedian]]s of a quadrilateral are the line segments connecting the [[midpoint]]s of the opposite sides. The intersection of the bimedians is the [[centroid]] of the vertices of the quadrilateral.<ref name=":1" />

The midpoints of the sides of any quadrilateral (convex, concave or crossed) are the vertices of a [[parallelogram]] called the [[Varignon's theorem|Varignon parallelogram]]. It 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>[[Harold Scott MacDonald Coxeter|H. S. M. Coxeter]] and [[Samuel L. Greitzer|S. L. Greitzer]], Geometry Revisited, MAA, 1967, pp.&nbsp;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}}

In a convex quadrilateral with sides {{mvar|a}}, {{mvar|b}}, {{mvar|c}} and {{mvar|d}}, the length of the bimedian that connects the midpoints of the sides {{mvar|a}} and {{mvar|c}} is
:<math>m=\tfrac{1}{2}\sqrt{-a^2+b^2-c^2+d^2+p^2+q^2}</math>

where {{mvar|p}} and {{mvar|q}} are the length of the diagonals.<ref>{{cite web| url = http://www.artofproblemsolving.com/Forum/viewtopic.php?t=363253| title = Mateescu Constantin, Answer to ''Inequality Of Diagonal''| access-date = 2011-09-26| archive-date = 2014-10-24| archive-url = https://web.archive.org/web/20141024134419/http://www.artofproblemsolving.com/Forum/viewtopic.php?t=363253| url-status = dead}}</ref> The length of the bimedian that connects the midpoints of the sides {{mvar|b}} and {{mvar|d}} is
:<math>n=\tfrac{1}{2}\sqrt{a^2-b^2+c^2-d^2+p^2+q^2}.</math>

Hence<ref name=Altshiller-Court/>{{rp|p.126}}
:<math>\displaystyle p^2+q^2=2(m^2+n^2).</math>

This is also a [[corollary]] to the [[parallelogram law]] applied in the Varignon parallelogram.

The lengths of the bimedians can also be expressed in terms of two opposite sides and the distance {{mvar|x}} between the midpoints of the diagonals. This is possible when using Euler's quadrilateral theorem in the above formulas. Whence<ref name=Josefsson3/>
:<math>m=\tfrac{1}{2}\sqrt{2(b^2+d^2)-4x^2}</math>

and
:<math>n=\tfrac{1}{2}\sqrt{2(a^2+c^2)-4x^2}.</math>

Note that the two opposite sides in these formulas are not the two that the bimedian connects.

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
 | title = Characterizations of Orthodiagonal Quadrilaterals
 | url = http://forumgeom.fau.edu/FG2012volume12/FG201202.pdf
 | volume = 12
 | year = 2012
 | access-date = 2012-04-08
 | 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.