View source for Quadrilateral

{{Short description|Four-sided polygon}}
{{About| four-sided mathematical shapes}}
{{Redirect|Tetragon|the edible plant| Tetragonia tetragonioides}}
{{Infobox Polygon
| name          = Quadrilateral
| image         = Six Quadrilaterals.svg
| caption       = Some types of quadrilaterals
| edges         = 4
| schläfli      = {4} (for square)
| area          = various methods;<br />[[#Area of a convex quadrilateral|see below]]
| angle         = 90° (for square and rectangle)}}
In  [[Euclidean geometry|geometry]]  a '''quadrilateral''' is a four-sided [[polygon]], having four [[Edge (geometry)|edges]] (sides) and four [[Vertex (geometry)|corners]] (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a '''tetragon''', derived from Greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons (e.g. [[pentagon]]). Since "gon" means "angle", it is analogously called a '''quadrangle''', or 4-angle. A quadrilateral with vertices <math>A</math>, <math>B</math>, <math>C</math> and <math>D</math> is sometimes denoted as <math>\square ABCD</math>.<ref name=":0">{{Cite web|title=Quadrilaterals - Square, Rectangle, Rhombus, Trapezoid, Parallelogram|url=https://www.mathsisfun.com/quadrilaterals.html|access-date=2020-09-02|website=Mathsisfun.com}}</ref>

Quadrilaterals are either [[simple polygon|simple]] (not self-intersecting), or [[complex polygon|complex]] (self-intersecting, or crossed). Simple quadrilaterals are either [[convex polygon|convex]] or [[concave polygon|concave]].

The [[Internal and external angle|interior angles]] of a simple (and [[Plane (geometry)|planar]]) quadrilateral ''ABCD'' add up to 360 [[Degree (angle)|degrees]], that is<ref name=":0" />
:<math>\angle A+\angle B+\angle C+\angle D=360^{\circ}.</math>

This is a special case of the ''n''-gon interior angle sum formula: ''S'' = (''n'' − 2) × 180° (here, n=4).<ref>{{Cite web|url=https://www.cuemath.com/geometry/sum-of-angles-in-a-polygon/|title=Sum of Angles in a Polygon|website=Cuemath|access-date=22 June 2022}}</ref>

All non-self-crossing quadrilaterals [[tessellation|tile the plane]], by repeated rotation around the midpoints of their edges.<ref>{{citation|last=Martin|first=George Edward|doi=10.1007/978-1-4612-5680-9|isbn=0-387-90636-3|mr=718119|at=Theorem 12.1, page 120|publisher=Springer-Verlag|series=Undergraduate Texts in Mathematics|title=Transformation geometry|url=https://books.google.com/books?id=gevlBwAAQBAJ&pg=PA120|year=1982}}</ref>

==Simple quadrilaterals==
Any quadrilateral that is not self-intersecting is a simple quadrilateral.

===Convex quadrilateral===
[[File:Euler diagram of quadrilateral types.svg|thumb|300px|[[Euler diagram]] of some types of simple quadrilaterals. (UK) denotes British English and (US) denotes American English.]]
[[File:Symmetries_of_square.svg|300px|thumb|Convex quadrilaterals by symmetry, represented with a [[Hasse diagram]].]]

In a convex quadrilateral all interior angles are less than 180°, and the two diagonals both lie inside the quadrilateral.
*Irregular quadrilateral ([[British English]]) or trapezium ([[North American English]]): no sides are parallel.  (In British English, this was once called a ''trapezoid''. For more, see {{Section link|Trapezoid|Trapezium vs Trapezoid}}.)
*[[Trapezoid|Trapezium]] (UK) or [[trapezoid]] (US): at least one pair of opposite sides are [[parallel (geometry)|parallel]].  Trapezia (UK) and trapezoids (US) include parallelograms.
<!--Please do NOT define an isosceles trapezoid as having legs equal.  Doing so would make all parallelograms isosceles trapezoids, which we know is wrong.-->
*[[Isosceles trapezium]] (UK) or [[isosceles trapezoid]] (US): one pair of opposite sides are parallel and the base [[angle]]s are equal in measure. Alternative definitions are a quadrilateral with an axis of symmetry bisecting one pair of opposite sides, or a trapezoid with diagonals of equal length.
*[[Parallelogram]]: a quadrilateral with two pairs of parallel sides. Equivalent conditions are that opposite sides are of equal length; that opposite angles are equal; or that the diagonals bisect each other.  Parallelograms include rhombi (including those rectangles called squares) and rhomboids (including those rectangles called oblongs).  In other words, parallelograms include all rhombi and all rhomboids, and thus also include all rectangles.
*[[Rhombus]], rhomb:<ref name=":0" /> all four sides are of equal length (equilateral). An equivalent condition is that the diagonals perpendicularly bisect each other.  Informally: "a pushed-over square" (but strictly including a square, too).
*[[Rhomboid]]: a parallelogram in which adjacent sides are of unequal lengths, and some angles are [[Angle#Types of angles|oblique]] (equiv., having no right angles). Informally: "a pushed-over oblong". Not all references agree; some define a rhomboid as a parallelogram that is not a rhombus.<ref>{{cite web|url=http://www.cimt.plymouth.ac.uk/resources/topics/art002.pdf |title=Archived copy |access-date=June 20, 2013 |url-status=dead |archive-url=https://web.archive.org/web/20140514200449/http://www.cimt.plymouth.ac.uk/resources/topics/art002.pdf |archive-date=May 14, 2014 }}</ref>
*[[Rectangle]]: all four angles are right angles (equiangular). An equivalent condition is that the diagonals bisect each other, and are equal in length.  Rectangles include squares and oblongs. Informally: "a box or oblong" (including a square).
*[[Square (geometry)|Square]] (regular quadrilateral): all four sides are of equal length (equilateral), and all four angles are right angles. An equivalent condition is that opposite sides are parallel (a square is a parallelogram), and that the diagonals perpendicularly bisect each other and are of equal length. A quadrilateral is a square if and only if it is both a rhombus and a rectangle (i.e., four equal sides and four equal angles).
*[[wikt:oblong|Oblong]]: longer than wide, or wider than long (i.e., a rectangle that is not a square).<ref>{{Cite web|url=http://www.cleavebooks.co.uk/scol/calrect.htm|title=Rectangles Calculator|website=Cleavebooks.co.uk|access-date=1 March 2022}}</ref>
*[[Kite (geometry)|Kite]]: two pairs of adjacent sides are of equal length. This implies that one diagonal divides the kite into [[congruent triangles]], and so the angles between the two pairs of equal sides are equal in measure. It also implies that the diagonals are perpendicular. Kites include rhombi.
[[File:Quadrilaterals.svg]]
*[[Tangential quadrilateral]]: the four sides are tangents to an inscribed circle. A convex quadrilateral is tangential if and only if opposite sides have equal sums.
*[[Tangential trapezoid]]: a trapezoid where the four sides are [[tangent]]s to an [[inscribed circle]].
*[[Cyclic quadrilateral]]: the four vertices lie on a [[circumscribed circle]]. A convex quadrilateral is cyclic if and only if opposite angles sum to 180°.
*[[Right kite]]: a kite with two opposite right angles. It is a type of cyclic quadrilateral.
*[[Harmonic quadrilateral]]: a cyclic quadrilateral such that the products of the lengths of the opposing sides are equal.
*[[Bicentric quadrilateral]]: it is both tangential and cyclic.
*[[Orthodiagonal quadrilateral]]: the diagonals cross at [[right angle]]s.
*[[Equidiagonal quadrilateral]]: the diagonals are of equal length.
*Bisect-diagonal quadrilateral: one diagonal bisects the other into equal lengths. Every dart and kite is bisect-diagonal. When both diagonals bisect another, it's a parallelogram.  
*[[Ex-tangential quadrilateral]]: the four extensions of the sides are tangent to an [[excircle]].
*An ''equilic quadrilateral'' has two opposite equal sides that when extended, meet at 60°.
*A ''Watt quadrilateral'' is a quadrilateral with a pair of opposite sides of equal length.<ref>{{cite journal |first1=G. |last1=Keady |first2=P. |last2=Scales |first3=S. Z. |last3=Németh |title=Watt Linkages and Quadrilaterals |journal=[[The Mathematical Gazette]] |volume=88 |issue=513 |year=2004 |pages=475–492 |doi=10.1017/S0025557200176107 |s2cid=125102050 |url=http://www.m-a.org.uk/jsp/index.jsp?lnk=620 }}</ref>
*A ''quadric quadrilateral'' is a convex quadrilateral whose four vertices all lie on the perimeter of a square.<ref>{{cite journal |first=A. K. |last=Jobbings |title=Quadric Quadrilaterals |journal=The Mathematical Gazette |volume=81 |issue=491 |year=1997 |pages=220–224 |doi=10.2307/3619199 |jstor=3619199 |s2cid=250440553 }}</ref>
*A ''diametric quadrilateral'' is a cyclic quadrilateral having one of its sides as a diameter of the circumcircle.<ref>{{cite journal |first=R. A. |last=Beauregard |title=Diametric Quadrilaterals with Two Equal Sides |journal=College Mathematics Journal |volume=40 |issue=1 |year=2009 |pages=17–21 |doi=10.1080/07468342.2009.11922331 |s2cid=122206817 }}</ref>
*A ''Hjelmslev quadrilateral'' is a quadrilateral with two right angles at opposite vertices.<ref>{{cite book |first=R. |last=Hartshorne |title=Geometry: Euclid and Beyond |publisher=Springer |year=2005 |pages=429–430 |isbn=978-1-4419-3145-0 }}</ref>

===Concave quadrilaterals===
In a concave quadrilateral, one interior angle is bigger than 180°, and one of the two diagonals lies outside the quadrilateral.
*A ''dart'' (or arrowhead) is a [[Concave polygon|concave]] quadrilateral with bilateral symmetry like a kite, but where one interior angle is reflex. See [[kite (geometry)|Kite]].

==Complex quadrilaterals ==
[[File:DU21 facets.png|thumb|upright=0.8|An antiparallelogram]]
A [[list of self-intersecting polygons|self-intersecting]] quadrilateral is called variously a '''cross-quadrilateral''', '''crossed quadrilateral''', '''[[butterfly]] quadrilateral''' or '''[[bow-tie]] quadrilateral'''. In a crossed quadrilateral, the four "interior" angles on either side of the crossing (two [[acute angle|acute]] and two [[reflex angle|reflex]], all on the left or all on the right as the figure is traced out) add up to 720°.<ref>{{cite web|url=http://mysite.mweb.co.za/residents/profmd/stars.pdf|title=Stars: A Second Look|website=Mysite.mweb.co.za|access-date=March 1, 2022|archive-date=March 3, 2016|archive-url=https://web.archive.org/web/20160303182521/http://mysite.mweb.co.za/residents/profmd/stars.pdf|url-status=dead}}</ref>

*[[Isosceles trapezoid#Self-intersections|Crossed trapezoid]] (US) or trapezium (Commonwealth):<ref>{{cite web | url=https://blogs.adelaide.edu.au/maths-learning/2016/04/06/the-crossed-trapezium/ | title=The crossed trapezium | last=Butler | first=David | date=2016-04-06 | website=Making Your Own Sense | access-date=2017-09-13}}</ref> a crossed quadrilateral in which one pair of nonadjacent sides is parallel (like a [[trapezoid]]).
*[[Antiparallelogram]]: a crossed quadrilateral in which each pair of nonadjacent sides have equal lengths (like a [[parallelogram]]).
*[[Crossed rectangle]]: an antiparallelogram whose sides are two opposite sides and the two diagonals of a [[rectangle]], hence having one pair of parallel opposite sides.
*[[Square#Crossed square|Crossed square]]: a special case of a crossed rectangle where two of the sides intersect at right angles.

==Special line segments==
The two [[diagonal]]s of a convex quadrilateral are the [[line segment]]s that connect opposite vertices.

The two '''bimedians''' of a convex quadrilateral are the line segments that connect the midpoints of opposite sides.<ref>{{cite web |author=E.W. Weisstein |title=Bimedian |url=http://mathworld.wolfram.com/Bimedian.html |publisher=MathWorld – A Wolfram Web Resource}}</ref> They intersect at the "vertex centroid" of the quadrilateral (see {{Section link||Remarkable points and lines in a convex quadrilateral}} below).

The four '''maltitudes''' of a convex quadrilateral are the perpendiculars to a side—through the midpoint of the opposite side.<ref>{{cite web |author=E.W. Weisstein |title=Maltitude |url=http://mathworld.wolfram.com/Maltitude.html |publisher=MathWorld – A Wolfram Web Resource}}</ref>

==Area of a convex quadrilateral==
There are various general formulas for the [[area]] {{math|''K''}} of a convex quadrilateral ''ABCD'' with sides {{math|''a'' {{=}}  ''AB'', ''b'' {{=}} ''BC'', ''c'' {{=}} ''CD'' and ''d'' {{=}} ''DA''}}.

===Trigonometric formulas===
The area can be expressed in trigonometric terms as<ref name=":1">{{Cite web|last=Weisstein|first=Eric W.|title=Quadrilateral|url=https://mathworld.wolfram.com/Quadrilateral.html|access-date=2020-09-02|website=mathworld.wolfram.com|language=en}}</ref>
:<math>K = \tfrac12 pq \sin \theta,</math>

where the lengths of the diagonals are {{math|''p''}} and {{math|''q''}} and the angle between them is {{math|''θ''}}.<ref>Harries, J. "Area of a quadrilateral," ''Mathematical Gazette'' 86, July 2002, 310–311.</ref> In the case of an orthodiagonal quadrilateral (e.g. rhombus, square, and kite), this formula reduces to <math>K=\tfrac{pq}{2}</math> since {{math|''θ''}} is {{math|90°}}.

The area can be also expressed in terms of bimedians as<ref name=Josefsson4/>
:<math>K = mn \sin \varphi,</math>
where the lengths of the bimedians are {{math|''m''}} and {{math|''n''}} and the angle between them is {{math|''φ''}}.

[[Bretschneider's formula]]<ref>R. A. Johnson, ''Advanced Euclidean Geometry'', 2007, [[Dover Publications|Dover Publ.]], p. 82.</ref><ref name=":1" /> expresses the area in terms of the sides and two opposite angles:
:<math>\begin{align}
K &= \sqrt{(s-a)(s-b)(s-c)(s-d) - \tfrac{1}{2} abcd \; [ 1 + \cos (A + C) ]} \\
&= \sqrt{(s-a)(s-b)(s-c)(s-d) - abcd\, \cos^2 \tfrac12(A + C) }
\end{align}</math>

where the sides in sequence are {{math|''a''}}, {{math|''b''}}, {{math|''c''}}, {{math|''d''}}, where {{math|''s''}} is the semiperimeter, and {{math|''A''}} and {{math|''C''}} are two (in fact, any two) opposite angles. This reduces to [[Brahmagupta's formula]] for the area of a cyclic quadrilateral—when {{math|{{nobreak|''A'' + ''C'' {{=}} 180°}} }}.

Another area formula in terms of the sides and angles, with angle {{math|''C''}} being between sides {{math|''b''}} and {{math|''c''}}, and {{math|''A''}} being between sides {{math|''a''}} and {{math|''d''}}, is
:<math>K = \tfrac12 ad \sin{A} + \tfrac12 bc \sin{C}.</math>

In the case of a cyclic quadrilateral, the latter formula becomes <math>K = \tfrac12(ad+bc)\sin{A}.</math>

In a parallelogram, where both pairs of opposite sides and angles are equal, this formula reduces to <math>K=ab \cdot \sin{A}.</math>

Alternatively, we can write the area in terms of the sides and the intersection angle {{math|''θ''}} of the diagonals, as long as {{math|''θ''}} is not {{math|90°}}:<ref name=Mitchell>Mitchell, Douglas W., "The area of a quadrilateral," ''Mathematical Gazette'' 93, July 2009, 306–309.</ref>
:<math>K = \tfrac14 \left|\tan \theta\right| \cdot \left| a^2 + c^2 - b^2 - d^2 \right|.</math>

In the case of a parallelogram, the latter formula becomes <math>K = \tfrac12 \left|\tan \theta\right| \cdot \left| a^2 - b^2 \right|.</math>

Another area formula including the sides {{math|''a''}}, {{math|''b''}}, {{math|''c''}}, {{math|''d''}} is<ref name=Josefsson4>{{citation
 | last = Josefsson
 | first = Martin
 | journal = Forum Geometricorum
 | pages = 17–21
 | title = Five Proofs of an Area Characterization of Rectangles
 | url = http://forumgeom.fau.edu/FG2013volume13/FG201304.pdf
 | volume = 13
 | year = 2013
 | access-date = 2013-02-20
 | archive-date = 2016-03-04
 | archive-url = https://web.archive.org/web/20160304001152/http://forumgeom.fau.edu/FG2013volume13/FG201304.pdf
 | url-status = dead
 }}.</ref>
:<math>K=\tfrac12 \sqrt{\bigl((a^2+c^2)-2x^2\bigr)\bigl((b^2+d^2)-2x^2\bigr)} \sin{\varphi}</math>
where {{math|''x''}} is the distance between the midpoints of the diagonals, and {{math|''φ''}} is the angle between the [[Quadrilateral#Special line segments|bimedian]]s.

The last trigonometric area formula including the sides {{math|''a''}}, {{math|''b''}}, {{math|''c''}}, {{math|''d''}} and the angle {{math|''α''}} (between {{math|''a''}} and {{math|''b''}}) is:<ref>{{Cite web |date=2009 |title=Triangle formulae |url=https://www.mathcentre.ac.uk/resources/uploaded/mc-ty-triangleformulae-2009-1.pdf |access-date=26 June 2023 |website=mathcentre.ac.uk}}</ref>
:<math>K=\tfrac12 ab \sin{\alpha}+\tfrac14 \sqrt{4c^2d^2-(c^2+d^2-a^2-b^2+2ab \cos{\alpha})^2} ,</math>
which can also be used for the area of a concave quadrilateral (having the concave part opposite to angle {{math|''α''}}), by just changing the first sign {{math|+}} to {{math|-}}.

===Non-trigonometric formulas===
The following two formulas express the area in terms of the sides {{math|''a''}}, {{math|''b''}}, {{math|''c''}} and {{math|''d''}}, the [[Semiperimeter#Quadrilaterals|semiperimeter]] {{math|''s''}}, and the diagonals {{math|''p''}}, {{math|''q''}}:

:<math>K = \sqrt{(s-a)(s-b)(s-c)(s-d) - \tfrac{1}{4}(ac+bd+pq)(ac+bd-pq)},</math> <ref>J. L. Coolidge, "A historically interesting formula for the area of a quadrilateral", ''American Mathematical Monthly'', 46 (1939) 345–347.</ref>

:<math>K = \tfrac14 \sqrt{4p^2q^2 - \left( a^2 + c^2 - b^2 - d^2 \right)^2}.</math> <ref>{{cite web |author=E.W. Weisstein |title=Bretschneider's formula |url=http://mathworld.wolfram.com/BretschneidersFormula.html |publisher=MathWorld – A Wolfram Web Resource}}</ref>

The first reduces to Brahmagupta's formula in the cyclic quadrilateral case, since then {{math|1=''pq'' = ''ac'' + ''bd''}}.

The area can also be expressed in terms of the bimedians {{math|''m''}}, {{math|''n''}} and the diagonals {{math|''p''}}, {{math|''q''}}:

:<math>K=\tfrac12 \sqrt{(m+n+p)(m+n-p)(m+n+q)(m+n-q)},</math> <ref>Archibald, R. C., "The Area of a Quadrilateral", ''American Mathematical Monthly'', 29 (1922) pp. 29–36.</ref>

:<math>K=\tfrac12 \sqrt{p^2q^2-(m^2-n^2)^2}.</math> <ref name=Josefsson3>{{citation
 | last = Josefsson
 | first = Martin
 | journal = Forum Geometricorum
 | pages = 155–164
 | title = The Area of a Bicentric Quadrilateral
 | url = http://forumgeom.fau.edu/FG2011volume11/FG201116.pdf
 | volume = 11
 | year = 2011
 | access-date = 2012-02-08
 | archive-date = 2020-01-05
 | archive-url = https://web.archive.org/web/20200105031952/http://forumgeom.fau.edu/FG2011volume11/FG201116.pdf
 | url-status = dead
 }}.</ref>{{rp|Thm. 7}}

In fact, any three of the four values {{math|''m''}}, {{math|''n''}}, {{math|''p''}}, and {{math|''q''}} suffice for determination of the area, since in any quadrilateral the four values are related by <math>p^2+q^2=2(m^2+n^2).</math><ref name=Altshiller-Court/>{{rp|p. 126}} The corresponding expressions are:<ref name=Josefsson6>Josefsson, Martin (2016) ‘100.31 Heron-like formulas for quadrilaterals’, ''The Mathematical Gazette'', '''100''' (549), pp. 505–508.</ref>

:<math>K=\tfrac12 \sqrt{[(m+n)^2-p^2]\cdot[p^2-(m-n)^2]},</math> 
if the lengths of two bimedians and one diagonal are given, and<ref name=Josefsson6/>

:<math>K=\tfrac14 \sqrt{[(p+q)^2-4m^2]\cdot[4m^2-(p-q)^2]},</math> 
if the lengths of two diagonals and one bimedian are given.

===Vector formulas===
The area of a quadrilateral {{math|''ABCD''}} can be calculated using [[Vector (geometric)|vectors]]. Let vectors {{math|'''AC'''}} and {{math|'''BD'''}} form the diagonals from {{math|''A''}} to {{math|''C''}} and from {{math|''B''}} to {{math|''D''}}. The area of the quadrilateral is then
:<math>K = \tfrac12 |\mathbf{AC}\times\mathbf{BD}|,</math>

which is half the magnitude of the [[cross product]] of vectors {{math|'''AC'''}} and {{math|'''BD'''}}. In two-dimensional Euclidean space, expressing vector {{math|'''AC'''}} as a [[Euclidean vector#In Cartesian space|free vector in Cartesian space]] equal to {{math|('''''x''<sub>1</sub>,''y''<sub>1</sub>''')}} and {{math|'''BD'''}} as {{math|('''''x''<sub>2</sub>,''y''<sub>2</sub>''')}}, this can be rewritten as:
:<math>K = \tfrac12 |x_1 y_2 - x_2 y_1|.</math>

==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>

==Angle bisectors==
The internal [[angle bisector]]s of a convex quadrilateral either form a [[cyclic quadrilateral]]<ref name=Altshiller-Court/>{{rp|p.127}} (that is, the four intersection points of adjacent angle bisectors are [[concyclic points|concyclic]]) or they are [[Concurrent lines|concurrent]]. In the latter case the quadrilateral is a [[tangential quadrilateral]].

In quadrilateral ''ABCD'', if the [[bisection#Of angles|angle bisectors]] of {{mvar|A}} and {{mvar|C}} meet on diagonal {{mvar|BD}}, then the angle bisectors of {{mvar|B}} and {{mvar|D}} meet on diagonal {{mvar|AC}}.<ref>Leversha, Gerry, "A property of the diagonals of a cyclic quadrilateral", ''Mathematical Gazette'' 93, March 2009, 116–118.</ref>

==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.

==Trigonometric identities==
The four angles of a simple quadrilateral ''ABCD'' satisfy the following identities:<ref>C. V. Durell & A. Robson, ''Advanced Trigonometry'', Dover, 2003, p. 267.</ref>

:<math>\sin A + \sin B + \sin C + \sin D = 4\sin\tfrac12(A+B)\, \sin\tfrac12(A+C)\, \sin\tfrac12(A+D)</math>

and

:<math>
\frac{\tan A\,\tan{B} - \tan C\,\tan D}{\tan A\,\tan C - \tan B\,\tan D}
= \frac{\tan(A+C)}{\tan(A+B)}.
</math>

Also,<ref>{{cite web|url=http://www.mathpropress.com/archive/RabinowitzProblems1963-2005.pdf|title=Original Problems Proposed by Stanley Rabinowitz 1963–2005|website=Mathpropress.com|access-date=March 1, 2022}}</ref>

:<math>
\frac{\tan A + \tan B + \tan C + \tan D}{\cot A + \cot B + \cot C + \cot D}
= \tan{A}\tan{B}\tan{C}\tan{D}.
</math>

In the last two formulas, no angle is allowed to be a [[right angle]], since tan&nbsp;90° is not defined.

Let <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math> be the sides of a convex quadrilateral, <math>s</math> is the semiperimeter, 
and  <math>A</math> and <math>C</math> are opposite angles, then<ref>{{Cite web|url=http://matinf.upit.ro/MATINF6/index.html#p=1|title=E. A. José García, Two Identities and their Consequences, MATINF, 6 (2020) 5-11|website=Matinf.upit.ro|access-date=1 March 2022}}</ref>

:<math>ad\sin^2\tfrac12 A + bc\cos^2\tfrac12C = (s-a)(s-d)</math>

and

:<math>bc\sin^2\tfrac12 C + ad\cos^2\tfrac12 A = (s-b)(s-c)</math>.

We can use these identities to derive the [[Bretschneider's Formula]].

==Inequalities==
===Area===
If a convex quadrilateral has the consecutive sides ''a'', ''b'', ''c'', ''d'' and the diagonals ''p'', ''q'', then its area ''K'' satisfies<ref>O. Bottema, ''Geometric Inequalities'', Wolters–Noordhoff Publishing, The Netherlands, 1969, pp. 129, 132.</ref>
:<math>K\le \tfrac{1}{4}(a+c)(b+d)</math> with equality only for a [[rectangle]].
:<math>K\le \tfrac{1}{4}(a^2+b^2+c^2+d^2)</math> with equality only for a [[square]].
:<math>K\le \tfrac{1}{4}(p^2+q^2)</math> with equality only if the diagonals are perpendicular and equal.
:<math>K\le \tfrac{1}{2}\sqrt{(a^2+c^2)(b^2+d^2)}</math> with equality only for a rectangle.<ref name=Josefsson4/>

From [[Bretschneider's formula]] it directly follows that the area of a quadrilateral satisfies
:<math>K \le \sqrt{(s-a)(s-b)(s-c)(s-d)}</math>

with equality [[if and only if]] the quadrilateral is [[cyclic quadrilateral|cyclic]] or degenerate such that one side is equal to the sum of the other three (it has collapsed into a [[line segment]], so the area is zero).

Also,
:<math>K \leq \sqrt{abcd},</math>
with equality for a [[bicentric quadrilateral]] or a rectangle.

The area of any quadrilateral also satisfies the inequality<ref name=Alsina>{{citation|last1=Alsina|first1=Claudi|last2=Nelsen|first2=Roger|title=When Less is More: Visualizing Basic Inequalities|publisher=Mathematical Association of America|year=2009|page=68}}.</ref>
:<math>\displaystyle K\le \tfrac{1}{2}\sqrt[3]{(ab+cd)(ac+bd)(ad+bc)}.</math>

Denoting the perimeter as ''L'', we have<ref name=Alsina/>{{rp|p.114}}
:<math>K\le \tfrac{1}{16}L^2,</math>

with equality only in the case of a square.

The area of a convex quadrilateral also satisfies
:<math>K \le \tfrac{1}{2}pq</math>

for diagonal lengths ''p'' and ''q'', with equality if and only if the diagonals are perpendicular.

Let ''a'', ''b'', ''c'', ''d'' be the lengths of the sides of a convex quadrilateral ''ABCD'' with the area ''K'' and diagonals ''AC = p'', ''BD = q''. Then<ref>Dao Thanh Oai, Leonard Giugiuc, Problem 12033, American Mathematical Monthly, March 2018, p. 277</ref>

:<math> K \leq \tfrac18(a^2+b^2+c^2+d^2+p^2+q^2+pq-ac-bd) </math> with equality only for a square.

Let ''a'', ''b'', ''c'', ''d'' be the lengths of the sides of a convex quadrilateral ''ABCD'' with the area ''K'', then the following inequality holds:<ref>{{cite journal|author1=Leonard Mihai Giugiuc|author2=Dao Thanh Oai|author3=Kadir Altintas|title=An inequality related to the lengths and area of a convex quadrilateral|journal=International Journal of Geometry|volume=7|date=2018|pages=81–86|url=https://ijgeometry.com/wp-content/uploads/2018/04/81-86.pdf}}</ref>

:<math> K \leq \frac{1}{3+\sqrt{3}}(ab+ac+ad+bc+bd+cd) - \frac{1}{2(1+\sqrt{3})^2}(a^2+b^2+c^2+d^2) </math> with equality only for a square.

===Diagonals and bimedians===
A corollary to Euler's quadrilateral theorem is the inequality
:<math> a^2 + b^2 + c^2 + d^2 \ge p^2 + q^2 </math>

where equality holds if and only if the quadrilateral is a [[parallelogram]].

[[Leonhard Euler|Euler]] also generalized [[Ptolemy's theorem]], which is an equality in a [[cyclic quadrilateral]], into an inequality for a convex quadrilateral. It states that
:<math> pq \le ac + bd </math>

where there is equality [[if and only if]] the quadrilateral is cyclic.<ref name=Altshiller-Court/>{{rp|p.128–129}} This is often called [[Ptolemy's inequality]].

In any convex quadrilateral the bimedians ''m, n'' and the diagonals ''p, q'' are related by the inequality
:<math>pq \leq m^2+n^2,</math>

with equality holding if and only if the diagonals are equal.<ref name=J2014>{{cite journal |last=Josefsson |first=Martin |title=Properties of equidiagonal quadrilaterals |journal=Forum Geometricorum |volume=14 |year=2014 |pages=129–144 |url=http://forumgeom.fau.edu/FG2014volume14/FG201412index.html |access-date=2014-08-28 |archive-date=2024-06-05 |archive-url=https://web.archive.org/web/20240605032351/https://forumgeom.fau.edu/FG2014volume14/FG201412index.html |url-status=dead }}</ref>{{rp|Prop.1}} This follows directly from the quadrilateral identity <math>m^2+n^2=\tfrac{1}{2}(p^2+q^2).</math>

===Sides===
The sides ''a'', ''b'', ''c'', and ''d'' of any quadrilateral satisfy<ref name=Crux>{{cite web|url=http://www.imomath.com/othercomp/Journ/ineq.pdf|title=Inequalities proposed in ''Crux Mathematicorum'' (from vol. 1, no. 1 to vol. 4, no. 2 known as "Eureka")|website=Imomath.com|access-date=March 1, 2022}}</ref>{{rp|p.228,#275}}
:<math>a^2+b^2+c^2 > \tfrac13 d^2</math>

and<ref name=Crux/>{{rp|p.234,#466}}
:<math>a^4+b^4+c^4 \geq \tfrac1{27} d^4.</math>

==Maximum and minimum properties==
Among all quadrilaterals with a given [[perimeter]], the one with the largest area is the [[Square (geometry)|square]]. This is called the ''[[isoperimetric inequality|isoperimetric theorem]] for quadrilaterals''. It is a direct consequence of the area inequality<ref name=Alsina/>{{rp|p.114}}
:<math>K\le \tfrac{1}{16}L^2</math>

where ''K'' is the area of a convex quadrilateral with perimeter ''L''. Equality holds [[if and only if]] the quadrilateral is a square. The dual theorem states that of all quadrilaterals with a given area, the square has the shortest perimeter.

The quadrilateral with given side lengths that has the [[Maxima and minima|maximum]] area is the [[cyclic quadrilateral]].<ref name=Peter/>

Of all convex quadrilaterals with given diagonals, the [[orthodiagonal quadrilateral]] has the largest area.<ref name=Alsina/>{{rp|p.119}} This is a direct consequence of the fact that the area of a convex quadrilateral satisfies
:<math>K=\tfrac{1}{2}pq\sin{\theta}\le \tfrac{1}{2}pq,</math>

where ''θ'' is the angle between the diagonals ''p'' and ''q''. Equality holds if and only if ''θ'' = 90°.

If ''P'' is an interior point in a convex quadrilateral ''ABCD'', then
:<math>AP+BP+CP+DP\ge AC+BD.</math>

From this inequality it follows that the point inside a quadrilateral that [[Maxima and minima|minimizes]] the sum of distances to the [[Vertex (geometry)|vertices]] is the intersection of the diagonals. Hence that point is the [[Fermat point]] of a convex quadrilateral.<ref name=autogenerated1>{{cite book |last1=Alsina |first1=Claudi |last2=Nelsen |first2=Roger |title=Charming Proofs : A Journey Into Elegant Mathematics |publisher=Mathematical Association of America |year=2010 |pages=114, 119, 120, 261 |isbn=978-0-88385-348-1 }}</ref>{{rp|p.120}}

==Remarkable points and lines in a convex quadrilateral==
The centre of a quadrilateral can be defined in several different ways. The "vertex centroid" comes from considering the quadrilateral as being empty but having equal masses at its vertices. The "side centroid" comes from considering the sides to have constant mass per unit length. The usual centre, called just [[centroid]] (centre of area) comes from considering the surface of the quadrilateral as having constant density. These three points are in general not all the same point.<ref>{{Cite web|url=https://sites.math.washington.edu/~king/java/gsp/center-mass-quad.html|title=Two Centers of Mass of a Quadrilateral|website=Sites.math.washington.edu|access-date=1 March 2022}}</ref>

The "vertex centroid" is the intersection of the two [[Quadrilateral#Special line segments|bimedians]].<ref>Honsberger, Ross, ''Episodes in Nineteenth and Twentieth Century Euclidean Geometry'', Math. Assoc. Amer., 1995, pp. 35–41.</ref> As with any polygon, the ''x'' and ''y'' coordinates of the vertex centroid are the [[arithmetic mean]]s of the ''x'' and ''y'' coordinates of the vertices.

The "area centroid" of quadrilateral ''ABCD'' can be constructed in the following way. Let ''G<sub>a</sub>'', ''G<sub>b</sub>'', ''G<sub>c</sub>'', ''G<sub>d</sub>'' be the centroids of triangles ''BCD'', ''ACD'', ''ABD'', ''ABC'' respectively. Then the "area centroid" is the intersection of the lines ''G<sub>a</sub>G<sub>c</sub>'' and ''G<sub>b</sub>G<sub>d</sub>''.<ref name=Myakishev>{{citation
 | last = Myakishev
 | first = Alexei
 | journal = Forum Geometricorum
 | pages = 289–295
 | title = On Two Remarkable Lines Related to a Quadrilateral
 | url = http://forumgeom.fau.edu/FG2006volume6/FG200634.pdf
 | volume = 6
 | year = 2006
 | access-date = 2012-04-15
 | archive-date = 2019-12-31
 | archive-url = https://web.archive.org/web/20191231055834/http://forumgeom.fau.edu/FG2006volume6/FG200634.pdf
 | url-status = dead
 }}.</ref>

In a general convex quadrilateral ''ABCD'', there are no natural analogies to the [[circumcenter]] and [[orthocenter]] of a [[triangle]]. But two such points can be constructed in the following way. Let ''O<sub>a</sub>'', ''O<sub>b</sub>'', ''O<sub>c</sub>'', ''O<sub>d</sub>'' be the circumcenters of triangles ''BCD'', ''ACD'', ''ABD'', ''ABC'' respectively; and denote by ''H<sub>a</sub>'', ''H<sub>b</sub>'', ''H<sub>c</sub>'', ''H<sub>d</sub>'' the orthocenters in the same triangles. Then the intersection of the lines ''O<sub>a</sub>O<sub>c</sub>'' and ''O<sub>b</sub>O<sub>d</sub>'' is called the [[circumcenter of mass|quasicircumcenter]], and the intersection of the lines ''H<sub>a</sub>H<sub>c</sub>'' and ''H<sub>b</sub>H<sub>d</sub>'' is called the ''quasiorthocenter'' of the convex quadrilateral.<ref name=Myakishev/> These points can be used to define an [[Euler line]] of a quadrilateral. In a convex quadrilateral, the quasiorthocenter ''H'', the "area centroid" ''G'', and the quasicircumcenter ''O'' are [[collinear]] in this order, and ''HG'' = 2''GO''.<ref name=Myakishev/>

There can also be defined a ''quasinine-point center'' ''E'' as the intersection of the lines ''E<sub>a</sub>E<sub>c</sub>'' and ''E<sub>b</sub>E<sub>d</sub>'', where ''E<sub>a</sub>'', ''E<sub>b</sub>'', ''E<sub>c</sub>'', ''E<sub>d</sub>'' are the [[Nine-point circle|nine-point centers]] of triangles ''BCD'', ''ACD'', ''ABD'', ''ABC'' respectively. Then ''E'' is the [[midpoint]] of ''OH''.<ref name=Myakishev/>

Another remarkable line in a convex non-parallelogram quadrilateral is the [[Newton line]], which connects the midpoints of the diagonals, the segment connecting these points being bisected by the vertex centroid. One more interesting line (in some sense dual to the [[Newton line|Newton's]] one) is the line connecting the point of intersection of diagonals with the vertex centroid. The line is remarkable by the fact that it contains the (area) centroid. The vertex centroid divides the segment connecting the intersection of diagonals and the (area) centroid in the ratio 3:1.<ref>{{cite web|url=https://www.austms.org.au/Publ/Gazette/2010/May10/TechPaperMiller.pdf|title=Centroid of a quadrilateral|author=John Boris Miller|website=Austmd.org.au|access-date=March 1, 2022}}</ref>

For any quadrilateral ''ABCD'' with points ''P'' and ''Q'' the intersections of ''AD'' and ''BC'' and ''AB'' and ''CD'', respectively, the circles ''(PAB), (PCD), (QAD),'' and ''(QBC)'' pass through a common point ''M'', called a Miquel point.<ref>{{Cite book|title=Euclidean Geometry in Mathematical Olympiads|last=Chen|first=Evan|author-link=Evan Chen|publisher=Mathematical Association of America|year=2016|isbn=9780883858394|location=Washington, D.C.|pages=198}}</ref>

For a convex quadrilateral ''ABCD'' in which ''E'' is the point of intersection of the diagonals and ''F'' is the point of intersection of the extensions of sides ''BC'' and ''AD'', let ω be a circle through ''E'' and ''F'' which meets ''CB'' internally at ''M'' and ''DA'' internally at ''N''. Let ''CA'' meet ω again at ''L'' and let ''DB'' meet ω again at ''K''. Then there holds: the straight lines ''NK'' and ''ML'' intersect at point ''P'' that is located on the side ''AB''; the straight lines ''NL'' and ''KM'' intersect at point ''Q'' that is located on the side ''CD''. 
Points ''P'' and ''Q'' are called "Pascal points" formed by circle ω on sides ''AB'' and ''CD''.
<ref name=Fraivert>{{citation
 | last = David
 | first = Fraivert
 | journal = [[The Mathematical Gazette]]
 | pages = 233–239
 | title = Pascal-points quadrilaterals inscribed in a cyclic quadrilateral
 | volume = 103
 | year = 2019| issue = 557
 | doi = 10.1017/mag.2019.54
 | s2cid = 233360695
 }}.</ref>
<ref name=Fraivert2>{{citation
 | last = David
 | first = Fraivert
 | journal = Journal for Geometry and Graphics
 | pages = 5–27
 | title = A Set of Rectangles Inscribed in an Orthodiagonal Quadrilateral and Defined by Pascal-Points Circles
 | url = http://www.heldermann.de/JGG/JGG23/JGG231/jgg23002.htm
 | volume = 23
 | year = 2019}}.</ref>
<ref name=Fraivert3>{{citation
 | last = David
 | first = Fraivert
 | journal = [[Forum Geometricorum]]
 | pages = 509–526
 | title = Properties of a Pascal points circle in a quadrilateral with perpendicular diagonals
 | url = http://forumgeom.fau.edu/FG2017volume17/FG201748.pdf
 | volume = 17
 | year = 2017
 | access-date = 2020-04-29
 | archive-date = 2020-12-05
 | archive-url = https://web.archive.org/web/20201205215507/http://forumgeom.fau.edu/FG2017volume17/FG201748.pdf
 | url-status = dead
 }}.</ref>

==Other properties of convex quadrilaterals==
* If exterior squares are drawn on all sides of a quadrilateral then the segments connecting the [[Centre (geometry)#Symmetric objects|centers]] of opposite squares are (a) equal in length, and (b) [[perpendicular]]. Thus these centers are the vertices of an [[orthodiagonal quadrilateral]]. This is called [[Van Aubel's theorem]].
*For any simple quadrilateral with given edge lengths, there is a [[cyclic quadrilateral]] with the same edge lengths.<ref name=Peter>Peter, Thomas, "Maximizing the Area of a Quadrilateral", ''The College Mathematics Journal'', Vol. 34, No. 4 (September 2003), pp. 315–316.</ref>
*The four smaller triangles formed by the diagonals and sides of a convex quadrilateral have the property that the product of the areas of two opposite triangles equals the product of the areas of the other two triangles.<ref>{{cite journal|author=Josefsson, Martin|url=http://forumgeom.fau.edu/FG2013volume13/FG201305.pdf|archive-url=https://web.archive.org/web/20130616225118/http://forumgeom.fau.edu/FG2013volume13/FG201305.pdf|url-status=dead|archive-date=June 16, 2013|title=Characterizations of Trapezoids|journal=Forum Geometricorum|volume=13|date=2013|pages=23–35}}</ref>
*The angle <math>\theta</math> at the intersection of the diagonals satisfies <math display=block>\cos \theta = \frac{a^2+c^2-b^2-d^2}{2pq},</math> where <math>p, q</math> are the diagonals of the quadrilateral.<ref>{{cite book |last1=Alsina |first1=Claudi |last2=Nelsen |first2=Roger |year=2020 |title=A Cornucopia of Quadrilaterals |publisher=American Mathematical Society |at=[https://books.google.com/books?id=CGDSDwAAQBAJ&dq=%22Adding%20the%20four%20equations%2C%20noting%22&pg=PA18 {{pgs|17–18}}] |isbn=978-1-47-045312-1 |url=https://books.google.com/books?id=CGDSDwAAQBAJ}}</ref>

==Taxonomy==
[[File:Quadrilateral hierarchy svg.svg|thumb|A taxonomy of quadrilaterals, using a [[Hasse diagram]].]]
A hierarchical [[Taxonomy (general)|taxonomy]] of quadrilaterals is illustrated by the figure to the right. Lower classes are special cases of higher classes they are connected to. Note that "trapezoid" here is referring to the North American definition (the British equivalent is a trapezium). Inclusive definitions are used throughout.

==Skew quadrilaterals==
{{See also|Skew polygon}}
[[File:Disphenoid tetrahedron.png|260px|thumb|The (red) side edges of [[tetragonal disphenoid]] represent a regular zig-zag skew quadrilateral]]

A non-planar quadrilateral is called a '''skew quadrilateral'''. Formulas to compute its dihedral angles from the edge lengths and the angle between two adjacent edges were derived for work on the properties of molecules such as [[cyclobutane]] that contain a "puckered" ring of four atoms.<ref>{{cite journal |first1=M. P. |last1=Barnett |first2=J. F. |last2=Capitani |title=Modular chemical geometry and symbolic calculation |journal=International Journal of Quantum Chemistry |volume=106 |issue=1 |pages=215–227 |year=2006 |doi=10.1002/qua.20807 |bibcode=2006IJQC..106..215B }}</ref> Historically the term '''gauche quadrilateral''' was also used to mean a skew quadrilateral.<ref>{{cite journal |last=Hamilton |first=William Rowan |url=http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Gauche/Gauche1.pdf |title=On Some Results Obtained by the Quaternion Analysis Respecting the Inscription of "Gauche" Polygons in Surfaces of the Second Order |journal=Proceedings of the Royal Irish Academy |volume=4 |year=1850 |pages=380–387 }}</ref> A skew  quadrilateral together with its diagonals form a (possibly non-regular) [[tetrahedron]], and conversely every skew quadrilateral comes from a tetrahedron where a pair of opposite [[edge (geometry)|edge]]s is removed.

==See also==
*[[Complete quadrangle]]
*[[Perpendicular bisector construction of a quadrilateral]]
*[[Saccheri quadrilateral]]
*{{Section link|Types of mesh|Quadrilateral}}
*[[Quadrangle (geography)]]
*[[Homography]] - Any quadrilateral can be transformed into another quadrilateral by a projective transformation (homography)

{{clear}}

==References==
{{Reflist}}

==External links==
{{Commons category|Tetragons}}
* {{springer|title=Quadrangle, complete|id=p/q076010}}
* [http://www.cut-the-knot.org/Curriculum/Geometry/PerpBisectQuadri.shtml Quadrilaterals Formed by Perpendicular Bisectors], [http://www.cut-the-knot.org/Curriculum/Geometry/ProjectiveQuadri.shtml Projective Collinearity] and  [http://www.cut-the-knot.org/Curriculum/Geometry/Quadrilaterals.shtml Interactive Classification] of Quadrilaterals from [[cut-the-knot]]
* [http://www.mathopenref.com/tocs/quadrilateraltoc.html Definitions and examples of quadrilaterals] and [http://www.mathopenref.com/tetragon.html Definition and properties of tetragons] from Mathopenref
* [http://dynamicmathematicslearning.com/quad-tree-new-web.html A (dynamic) Hierarchical Quadrilateral Tree] at [http://dynamicmathematicslearning.com/JavaGSPLinks.htm Dynamic Geometry Sketches]
* [http://mysite.mweb.co.za/residents/profmd/quadclassify.pdf An extended classification of quadrilaterals] {{Webarchive|url=https://web.archive.org/web/20191230004754/http://mysite.mweb.co.za/residents/profmd/quadclassify.pdf |date=2019-12-30 }} at [http://mysite.mweb.co.za/residents/profmd/homepage4.html Dynamic Math Learning Homepage] {{Webarchive|url=https://web.archive.org/web/20180825150046/http://mysite.mweb.co.za/residents/profmd/homepage4.html |date=2018-08-25 }}
* [https://web.archive.org/web/20110719175018/http://mzone.mweb.co.za/residents/profmd/classify.pdf The role and function of a hierarchical classification of quadrilaterals] by Michael de Villiers

{{Polygons}}

[[Category:4 (number)]]
[[Category:Quadrilaterals| ]]