Editing Quadrilateral (section)

==Diagonals==
===Properties of the diagonals in quadrilaterals===
In the following table it is listed if the diagonals in some of the most basic quadrilaterals bisect each other, if their diagonals are [[perpendicular]], and if their diagonals have [[congruence (geometry)|equal length]].<ref>{{Cite web|url=https://math.okstate.edu/geoset/Projects/Ideas/QuadDiags.htm|title=Diagonals of Quadrilaterals -- Perpendicular, Bisecting or Both|website=Math.okstate.edu|access-date=1 March 2022}}</ref> The list applies to the most general cases, and excludes named subsets.

{| class="wikitable"
|-
! scope="col" | Quadrilateral
! scope="col" | Bisecting diagonals
! scope="col" | Perpendicular diagonals
! scope="col" | Equal diagonals
|-
! scope="row" | [[Trapezoid]] 
| {{No}} || ''See note 1'' || {{No}}
|-
! scope="row" | [[Isosceles trapezoid]] 
| {{No}} || ''See note 1'' || {{Yes}}
|-<!--
! scope="row" | [[Right trapezoid]]
|| ''See note 3'' || ''See note 1'' || {{No}}
|--->
! scope="row" | [[Parallelogram]] 
| {{Yes}} || {{No}} || {{No}}
|-
! scope="row" | [[Kite (geometry)|Kite]] 
| ''See note 2'' || {{Yes}} || ''See note 2''
|-
! scope="row" | [[Rectangle]] 
| {{Yes}} || {{No}} || {{Yes}}
|-
! scope="row" | [[Rhombus]] 
| {{Yes}} || {{Yes}} || {{No}}
|-
! scope="row" | [[Square]] 
| {{Yes}} || {{Yes}} || {{Yes}}
|}

* ''Note 1: The most general trapezoids and isosceles trapezoids do not have perpendicular diagonals, but there are infinite numbers of (non-similar) trapezoids and isosceles trapezoids that do have perpendicular diagonals and are not any other named quadrilateral.''
* ''Note 2: In a kite, one diagonal bisects the other.  The most general kite has unequal diagonals, but there is an infinite number of (non-similar) kites in which the diagonals are equal in length (and the kites are not any other named quadrilateral).''

===Lengths of the diagonals===
The lengths of the diagonals in a convex quadrilateral ''ABCD'' can be calculated using the [[law of cosines]] on each triangle formed by one diagonal and two sides of the quadrilateral. Thus
:<math>p=\sqrt{a^2+b^2-2ab\cos{B}}=\sqrt{c^2+d^2-2cd\cos{D}}</math>

and
:<math>q=\sqrt{a^2+d^2-2ad\cos{A}}=\sqrt{b^2+c^2-2bc\cos{C}}.</math>

Other, more symmetric formulas for the lengths of the diagonals, are<ref>Rashid, M. A. & Ajibade, A. O., "Two conditions for a quadrilateral to be cyclic expressed in terms of the lengths of its sides", ''Int. J. Math. Educ. Sci. Technol.'', vol. 34 (2003) no. 5, pp. 739–799.</ref>
:<math>p=\sqrt{\frac{(ac+bd)(ad+bc)-2abcd(\cos{B}+\cos{D})}{ab+cd}}</math>

and
:<math>q=\sqrt{\frac{(ab+cd)(ac+bd)-2abcd(\cos{A}+\cos{C})}{ad+bc}}.</math>

===Generalizations of the parallelogram law and Ptolemy's theorem===
In any convex quadrilateral ''ABCD'', the sum of the squares of the four sides is equal to the sum of the squares of the two diagonals plus four times the square of the line segment connecting the midpoints of the diagonals. Thus
:<math> a^2 + b^2 + c^2 + d^2 = p^2 + q^2 + 4x^2 </math>

where {{mvar|x}} is the distance between the midpoints of the diagonals.<ref name=Altshiller-Court/>{{rp|p.126}} This is sometimes known as [[Euler's quadrilateral theorem]] and is a generalization of the [[parallelogram law]].

The German mathematician [[Carl Anton Bretschneider]] derived in 1842 the following generalization of [[Ptolemy's theorem]], regarding the product of the diagonals in a convex quadrilateral<ref>Andreescu, Titu & Andrica, Dorian, ''Complex Numbers from A to...Z'', Birkhäuser, 2006, pp. 207–209.</ref>
:<math> p^2q^2=a^2c^2+b^2d^2-2abcd\cos{(A+C)}.</math>

This relation can be considered to be a [[law of cosines]] for a quadrilateral. In a [[cyclic quadrilateral]], where {{math|1=''A'' + ''C'' = 180°}}, it reduces to {{math|1=''pq'' = ''ac'' + ''bd''}}. Since {{math|cos{{thinsp}}(''A'' + ''C'') ≥ &minus;1}}, it also gives a proof of Ptolemy's inequality.

===Other metric relations===
If {{mvar|X}} and {{mvar|Y}} are the feet of the normals from {{mvar|B}} and {{mvar|D}} to the diagonal {{math|1=''AC'' = ''p''}} in a convex quadrilateral ''ABCD'' with sides {{math|1=''a'' = ''AB''}}, {{math|1=''b'' = ''BC''}}, {{math|1=''c'' = ''CD''}}, {{math|1=''d'' = ''DA''}}, then<ref name=Josefsson/>{{rp|p.14}}
:<math>XY=\frac{|a^2+c^2-b^2-d^2|}{2p}.</math>

In a convex quadrilateral ''ABCD'' with sides {{math|1=''a'' = ''AB''}}, {{math|1=''b'' = ''BC''}}, {{math|1=''c'' = ''CD''}}, {{math|1=''d'' = ''DA''}}, and where the diagonals intersect at {{mvar|E}},
:<math> efgh(a+c+b+d)(a+c-b-d) = (agh+cef+beh+dfg)(agh+cef-beh-dfg)</math>

where {{math|1=''e'' = ''AE''}}, {{math|1=''f'' = ''BE''}}, {{math|1=''g'' = ''CE''}}, and {{math|1=''h'' = ''DE''}}.<ref>{{citation
 | last = Hoehn
 | first = Larry
 | journal = Forum Geometricorum
 | pages = 211–212
 | title = A New Formula Concerning the Diagonals and Sides of a Quadrilateral
 | url = http://forumgeom.fau.edu/FG2011volume11/FG201122.pdf
 | volume = 11
 | year = 2011
 | access-date = 2012-04-28
 | archive-date = 2013-06-16
 | archive-url = https://web.archive.org/web/20130616232126/http://forumgeom.fau.edu/FG2011volume11/FG201122.pdf
 | url-status = dead
 }}.</ref>

The shape and size of a convex quadrilateral are fully determined by the lengths of its sides in sequence and of one diagonal between two specified vertices.  The two diagonals {{math|''p'', ''q''}} and the four side lengths {{math|''a'', ''b'', ''c'', ''d''}} of a quadrilateral are related<ref name=":1" /> by the [[Distance geometry#Cayley.E2.80.93Menger determinants|Cayley-Menger]] [[determinant]], as follows:
:<math> \det \begin{bmatrix} 
   0 & a^2 & p^2 & d^2 & 1 \\
 a^2 &   0 & b^2 & q^2 & 1 \\
 p^2 & b^2 &   0 & c^2 & 1 \\
 d^2 & q^2 & c^2 &   0 & 1 \\
   1 &   1 &   1 & 1   & 0
\end{bmatrix} = 0. </math>