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| {{About| four-sided mathematical shapes}}
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| {{Infobox Polygon
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| | name = Quadrilateral
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| | image = Six Quadrilaterals.svg
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| | caption = Six different types of quadrilaterals
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| | edges = 4
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| | schläfli = {4} (for square)
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| | area = various methods;<br>[[#Area of a convex quadrilateral|see below]]
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| | angle = 90° (for square and rectangle)}}
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| In [[Euclidean geometry|Euclidean plane geometry]], a '''quadrilateral''' is a [[polygon]] with four sides (or edges) and four vertices or corners. Sometimes, the term '''quadrangle''' is used, by analogy with [[triangle]], and sometimes '''tetragon''' for consistency with [[pentagon]] (5-sided), [[hexagon]] (6-sided) and so on.
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| The origin of the word "quadrilateral" is the two Latin words ''quadri'', a variant of four, and ''latus'', meaning "side."
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| Quadrilaterals are [[simple polygon|simple]] (not self-intersecting) or [[complex polygon|complex]] (self-intersecting), also called crossed. Simple quadrilaterals are either [[convex polygon|convex]] or [[concave polygon|concave]].
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| The [[Internal and external angle|interior angles]] of a simple (and planar) quadrilateral ''ABCD'' add up to 360 [[degrees of arc]], that is
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| :<math>\angle A+\angle B+\angle C+\angle D=360^{\circ}.</math>
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| This is a special case of the ''n''-gon interior angle sum formula (''n'' − 2) × 180°. In a crossed quadrilateral, the four interior angles on either side of the crossing add up to 720°.<ref>[http://mysite.mweb.co.za/residents/profmd/stars.pdf Stars: A Second Look]</ref>
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| All convex quadrilaterals [[tessellation|tile the plane]] by repeated rotation around the midpoints of their edges.
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| =={{anchor|Convex quadrilaterals - parallelograms}}Convex quadrilaterals – parallelograms==
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| [[File:Euler diagram of quadrilateral types.svg|thumb|300px|[[Euler diagram]] of some types of quadrilaterals. (UK) denotes British English and (US) denotes American English.]]
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| A [[parallelogram]] is 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 also include the square, rectangle, rhombus and rhomboid.
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| *[[Rhombus]] or rhomb: all four sides are of equal length. An equivalent condition is that the diagonals perpendicularly bisect each other. An informal description is "a pushed-over square" (including a square).
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| *[[Rhomboid]]: a parallelogram in which adjacent sides are of unequal lengths and angles are [[Angle#Types of angles|oblique]] (not right angles). Informally: "a pushed-over rectangle with no right angles."<ref>http://www.cimt.plymouth.ac.uk/resources/topics/art002.pdf</ref>
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| *[[Rectangle]]: all four angles are right angles. An equivalent condition is that the diagonals bisect each other and are equal in length. Informally: "a box or oblong" (including a square).
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| *[[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), 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 (four equal sides and four equal angles).
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| *[[Rectangle|Oblong]]: a term sometimes used to denote a rectangle which has unequal adjacent sides (i.e. a rectangle that is not a square).<ref>http://www.cleavebooks.co.uk/scol/calrect.htm</ref>
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| =={{anchor|Convex quadrilaterals - other}}Convex quadrilaterals – other==
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| *[[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.
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| *[[Right kite]]: a kite with two opposite right angles.
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| *[[Trapezoid]] ([[North American English]]) or [[trapezoid|Trapezium]] ([[British English]]): at least one pair of opposite sides are [[parallel (geometry)|parallel]].
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| *Trapezium (NAm.): no sides are parallel. (In British English this would be called an irregular quadrilateral, and was once called a ''trapezoid''.)
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| <!--Please do NOT define an isosceles trapezoid as having legs equal. Doing so would make all parallelograms isosceles trapezoids, which we know is wrong.-->
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| *[[Isosceles trapezoid]] (NAm.) or [[isosceles trapezium]] (Brit.): 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.
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| *[[Tangential trapezoid]]: a trapezoid where the four sides are [[tangent]]s to an [[inscribed circle]].
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| *[[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.
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| *[[Cyclic quadrilateral]]: the four vertices lie on a [[circumscribed circle]]. A convex quadrilateral is cyclic if and only if opposite angles sum to 180°.
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| *[[Bicentric quadrilateral]]: it is both tangential and cyclic.
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| *[[Orthodiagonal quadrilateral]]: the diagonals cross at [[right angle]]s.
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| *[[Equidiagonal quadrilateral]]: the diagonals are of equal length.
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| *[[Ex-tangential quadrilateral]]: the four extensions of the sides are tangent to an [[excircle]].
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| [[File:Quadrilaterals.svg]]
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| ==More quadrilaterals==
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| *An '''equilic quadrilateral''' has two opposite equal sides that, when extended, meet at 60°.
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| *A '''Watt quadrilateral''' is a quadrilateral with a pair of opposite sides of equal length.<ref>G. Keady, P. Scales and S. Z. Németh, "Watt Linkages and Quadrilaterals", ''The Mathematical Gazette'' Vol. 88, No. 513 (Nov., 2004), pp. 475–492.</ref>
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| *A '''quadric quadrilateral''' is a convex quadrilateral whose four vertices all lie on the perimeter of a square.<ref>A. K. Jobbings, "Quadric Quadrilaterals", ''The Mathematical Gazette'' Vol. 81, No. 491 (Jul., 1997), pp. 220–224.</ref>
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| *A geometric chevron (dart or arrowhead) is a [[Concave polygon|concave]] quadrilateral with bilateral symmetry like a kite, but one interior angle is reflex.
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| *A self-intersecting quadrilateral is called variously a '''cross-quadrilateral''', '''crossed quadrilateral''', '''[[butterfly]] quadrilateral''' or '''[[bow-tie]] quadrilateral'''. A special case of crossed quadrilaterals are the [[antiparallelogram]]s, crossed quadrilaterals in which (like a [[parallelogram]]) each pair of nonadjacent sides has equal length. The diagonals of a crossed or concave quadrilateral do not intersect inside the shape.
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| *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>M.P. Barnett and J.F. Capitani, ''Modular chemical geometry and symbolic calculation'',
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| International Journal of Quantum Chemistry, 106 (1) 215–227, 2006.</ref> See [[skew polygon]] for more. Historically the term '''gauche quadrilateral''' was also used to mean a skew quadrilateral.<ref>William Rowan Hamilton, ''[http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Gauche/Gauche1.pdf On Some Results Obtained by the Quaternion Analysis Respecting the Inscription of "Gauche" Polygons in Surfaces of the Second Order]'', Proceedings of the Royal Irish Academy, 4 (1850), pp. 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.
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| ==Special line segments==
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| The two ''diagonals'' of a convex quadrilateral are the [[line segment]]s that connect opposite vertices.
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| 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 [[Quadrilateral#Remarkable points and lines in a convex quadrilateral|Remarkable points]] below).
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| 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>
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| .
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| ==Area of a convex quadrilateral==
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| There are various general formulas for the [[area]] ''K'' of a convex quadrilateral.
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| ===Trigonometric formulas===
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| The area can be expressed in trigonometric terms as
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| :<math>K = \tfrac{1}{2} pq \cdot \sin \theta,</math>
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| where the lengths of the diagonals are ''p'' and ''q'' and the angle between them is ''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{1}{2}pq</math> since ''θ'' is 90°.
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| [[Bretschneider's formula]]<ref>R. A. Johnson, ''Advanced Euclidean Geometry'', 2007, Dover Publ., p. 82.</ref> expresses the area in terms of the sides and two opposite angles:
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| :<math>\begin{align}
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| K &= \sqrt{(s-a)(s-b)(s-c)(s-d) - \tfrac{1}{2} abcd \; [ 1 + \cos (A + C) ]} \\
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| &= \sqrt{(s-a)(s-b)(s-c)(s-d) - abcd \left[ \cos^2 \left( \tfrac{A + C}{2} \right) \right]} \\
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| \end{align}</math>
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| where the sides in sequence are ''a'', ''b'', ''c'', ''d'', where ''s'' is the semiperimeter, and ''A'' and ''C'' are two (in fact, any two) opposite angles. This reduces to [[Brahmagupta's formula]] for the area of a cyclic quadrilateral when ''A''+''C'' = 180°.
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| Another area formula in terms of the sides and angles, with angle ''C'' being between sides ''b'' and ''c'', and ''A'' being between sides ''a'' and ''d'', is
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| :<math>K = \tfrac{1}{2}ad \cdot \sin{A} + \tfrac{1}{2}bc \cdot \sin{C}.</math>
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| In the case of a cyclic quadrilateral, the latter formula becomes <math>K = \tfrac{1}{2}(ad+bc)\sin{A}.</math>
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| In a parallelogram, where both pairs of opposite sides and angles are equal, this formula reduces to <math>K=ab \cdot \sin{A}.</math>
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| Alternatively, we can write the area in terms of the sides and the intersection angle ''θ'' of the diagonals, so long as this angle is not 90°:<ref name=Mitchell>Mitchell, Douglas W., "The area of a quadrilateral," ''Mathematical Gazette'' 93, July 2009, 306–309.</ref>
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| :<math>K = \frac{|\tan \theta|}{4} \cdot \left| a^2 + c^2 - b^2 - d^2 \right|.</math>
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| In the case of a parallelogram, the latter formula becomes <math>K = \tfrac{1}{2}|\tan \theta|\cdot \left| a^2 - b^2 \right|.</math>
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| Another area formula including the sides ''a'', ''b'', ''c'', ''d'' is<ref name=Josefsson4>{{citation
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| | 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}}.</ref> | |
| :<math>K=\tfrac{1}{4}\sqrt{(2(a^2+c^2)-4x^2)(2(b^2+d^2)-4x^2)}\sin{\varphi}</math>
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| where ''x'' is the distance between the midpoints of the diagonals and ''φ'' is the angle between the [[Quadrilateral#Special_line_segments|bimedian]]s.
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| ===Non-trigonometric formulas===
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| The following two formulas expresses the area in terms of the sides ''a'', ''b'', ''c'', ''d'', the semiperimeter ''s'', and the diagonals ''p'', ''q'':
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| :<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>
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| :<math>K = \frac{1}{4} \sqrt{4p^{2}q^{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>
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| The first reduces to Brahmagupta's formula in the cyclic quadrilateral case, since then ''pq'' = ''ac'' + ''bd''.
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| The area can also be expressed in terms of the bimedians ''m'', ''n'' and the diagonals ''p'', ''q'':
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| :<math>K=\tfrac{1}{2}\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>
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| :<math>K=\tfrac{1}{2}\sqrt{p^2q^2-(m^2-n^2)^2}.</math> <ref name=Josefsson3>{{citation
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| | 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}}.</ref> | |
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| ===Vector formulas===
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| The area of a quadrilateral ''ABCD'' can be calculated using [[Vector (geometric)|vectors]]. Let vectors '''AC''' and '''BD''' form the diagonals from ''A'' to ''C'' and from ''B'' to ''D''. The area of the quadrilateral is then
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| :<math>K = \tfrac{1}{2} |\mathbf{AC}\times\mathbf{BD}|,</math>
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| which is half the magnitude of the [[cross product]] of vectors '''AC''' and '''BD'''. In two-dimensional Euclidean space, expressing vector '''AC''' as a [[Euclidean vector#In Cartesian space|free vector in Cartesian space]] equal to ('''''x''<sub>1</sub>,''y''<sub>1</sub>''') and '''BD''' as ('''''x''<sub>2</sub>,''y''<sub>2</sub>'''), this can be rewritten as:
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| :<math>K = \tfrac{1}{2} |x_1 y_2 - x_2 y_1|.</math>
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| ===Area inequalities===
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| 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>
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| :<math>K\le \tfrac{1}{4}(a+c)(b+d)</math> with equality only for a [[rectangle]].
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| :<math>K\le \tfrac{1}{4}(a^2+b^2+c^2+d^2)</math> with equality only for a [[Square (geometry)|square]].
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| :<math>K\le \tfrac{1}{4}(p^2+q^2)</math> with equality only if the diagonals are perpendicular and equal.
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| :<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/>
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| From [[Bretschneider's formula]] it directly follows that the area of a quadrilateral satisfies
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| :<math>K \le \sqrt{(s-a)(s-b)(s-c)(s-d)}</math>
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| 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).
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| 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>
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| :<math>\displaystyle K\le \tfrac{1}{2}\sqrt[3]{(ab+cd)(ac+bd)(ad+bc)}.</math>
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| ==Diagonals==
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| ===Properties of the diagonals in some quadrilaterals===
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| 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 equal length.<ref>Jennifer Kahle, Geometry: Basic ideas, [http://www.math.okstate.edu/geoset/Projects/Ideas/QuadDiags.htm], accessed 28 December 2012.</ref> The list applies to the most general cases, and excludes named subsets.
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| {| class="wikitable"
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| |-
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| ! Quadrilateral || Bisecting diagonals || Perpendicular diagonals || Equal diagonals
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| |-
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| ! [[Trapezoid]]
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| || No || ''See note 1'' || No
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| |-
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| ! [[Isosceles trapezoid]]
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| || No || ''See note 1'' || Yes
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| |-
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| ! [[Parallelogram]]
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| || Yes || No || No
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| |-
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| ! [[Kite (geometry)|Kite]]
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| || ''See note 2'' || Yes || ''See note 2''
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| |-
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| ! [[Rectangle]]
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| || Yes || No || Yes
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| |-
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| ! [[Rhombus]]
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| || Yes || Yes || No
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| |-
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| ! [[Square]]
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| || Yes || Yes || Yes
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| |}
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| ''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.''
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| ''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).''
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| ===Length of the diagonals===
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| The length of the diagonals in a convex quadrilateral ''ABCD'' can be calculated using the [[law of cosines]]. Thus
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| :<math>p=\sqrt{a^2+b^2-2ab\cos{B}}=\sqrt{c^2+d^2-2cd\cos{D}}</math>
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| and
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| :<math>q=\sqrt{a^2+d^2-2ad\cos{A}}=\sqrt{b^2+c^2-2bc\cos{C}}.</math>
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| Other, more symmetric formulas for the length 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>
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| :<math>p=\sqrt{\frac{(ac+bd)(ad+bc)-2abcd(\cos{B}+\cos{D})}{ab+cd}}</math>
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| and
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| :<math>q=\sqrt{\frac{(ab+cd)(ac+bd)-2abcd(\cos{A}+\cos{C})}{ad+bc}}.</math>
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| ===Generalizations of the parallelogram law and Ptolemy's theorem===
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| 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
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| :<math> a^2 + b^2 + c^2 + d^2 = p^2 + q^2 + 4x^2 </math>
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| where ''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]]. A corollary is the inequality
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| :<math> a^2 + b^2 + c^2 + d^2 \ge p^2 + q^2 </math>
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| where equality holds if and only if the quadrilateral is a [[parallelogram]].
| |
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| [[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
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| :<math> pq \le ac + bd </math>
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| 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''.
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| 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>
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| :<math> p^2q^2=a^2c^2+b^2d^2-2abcd\cos{(A+C)}.</math> | |
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| This relation can be considered to be a [[law of cosines]] for a quadrilateral. In a [[cyclic quadrilateral]], where ''A'' + ''C'' = 180°, it reduces to ''pq = ac + bd''. Since cos (''A'' + ''C'') ≥ -1, it also gives a proof of Ptolemy's inequality.
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| ===Other metric relations===
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| If ''X'' and ''Y'' are the feet of the normals from ''B'' and ''D'' to the diagonal ''AC'' = ''p'' in a convex quadrilateral ''ABCD'' with sides ''a'' = ''AB'', ''b'' = ''BC'', ''c'' = ''CD'', ''d'' = ''DA'', then<ref name=Josefsson/>{{rp|p.14}}
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| :<math>XY=\frac{|a^2+c^2-b^2-d^2|}{2p}.</math>
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| In a convex quadrilateral ''ABCD'' with sides ''a'' = ''AB'', ''b'' = ''BC'', ''c'' = ''CD'', ''d'' = ''DA'', and where the diagonals intersect at ''E'',
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| :<math> efgh(a+c+b+d)(a+c-b-d) = (agh+cef+beh+dfg)(agh+cef-beh-dfg)</math>
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| | |
| where ''e'' = ''AE'', ''f'' = ''BE'', ''g'' = ''CE'', and ''h'' = ''DE''.<ref>{{citation
| |
| | last = Hoehn | first = Larry
| |
| | journal = Forum Geometricorum
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| | 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}}.</ref> | |
| | |
| The shape of a convex quadrilateral is fully determined by the lengths of its sides in sequence and of one diagonal between two specified vertices. The two diagonals ''p, q'' and the four side lengths ''a, b, c, d'' of a quadrilateral are related<ref>{{cite web |author=E.W. Weisstein |title=Quadrilateral |url=http://mathworld.wolfram.com/Quadrilateral.html |publisher=''MathWorld'' - A Wolfram Web Resource}}</ref> by the [[Distance_geometry#Cayley.E2.80.93Menger_determinants|Cayley-Menger]] [[determinant]], as follows:
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| :<math> \det \begin{bmatrix}
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| 0 & a^2 & p^2 & d^2 & 1 \\
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| a^2 & 0 & b^2 & q^2 & 1 \\ | |
| p^2 & b^2 & 0 & c^2 & 1 \\
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| d^2 & q^2 & c^2 & 0 & 1 \\ | |
| 1 & 1 & 1 & 1 & 0
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| \end{bmatrix} = 0. </math>
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| ==Bimedians==
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| [[File:Varignon theorem convex.png|300px|thumb|The Varignon
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| parallelogram ''EFGH'']]
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| The [[midpoint]]s 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:
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| *Each pair of opposite sides of the Varignon parallelogram are parallel to a diagonal in the original quadrilateral.
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| *The length of a side in the Varignon parallelogram is half as long as the diagonal in the original quadrilateral it is parallel to.
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| *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>
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| *The [[perimeter]] of the Varignon parallelogram equals the sum of the diagonals of the original quadrilateral.
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| The diagonals of the Varignon parallelogram are the [[Quadrilateral#Special line segments|bimedian]]s of the original quadrilateral.
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| 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}}
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| 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
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| :<math>m=\tfrac{1}{2}\sqrt{-a^2+b^2-c^2+d^2+p^2+q^2}</math>
| |
| | |
| where ''p'' and ''q'' are the length of the diagonals.<ref>[http://www.artofproblemsolving.com/Forum/viewtopic.php?t=363253 Mateescu Constantin, Answer to ''Inequality Of Diagonal'']</ref> The length of the bimedian that connects the midpoints of the sides ''b'' and ''d'' is
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| :<math>n=\tfrac{1}{2}\sqrt{a^2-b^2+c^2-d^2+p^2+q^2}.</math>
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| Hence<ref name=Altshiller-Court/>{{rp|p.126}}
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| :<math>\displaystyle p^2+q^2=2(m^2+n^2).</math>
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| This is also a [[corollary]] to the [[parallelogram law]] applied in the Varignon parallelogram.
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| The length of the bimedians can also be expressed in terms of two opposite sides and the distance ''x'' between the midpoints of the diagonals. This is possible when using Euler's quadrilateral theorem in the above formulas. Whence<ref name=Josefsson3/>
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| :<math>m=\tfrac{1}{2}\sqrt{2(b^2+d^2)-4x^2}</math>
| |
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| and
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| :<math>n=\tfrac{1}{2}\sqrt{2(a^2+c^2)-4x^2}.</math>
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| Note that the two opposite sides in these formulas are not the two that the bimedian connects.
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| In a convex quadrilateral, there are 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}}.</ref> | |
| * The two bimedians have equal length [[if and only if]] the two diagonals are [[perpendicular]]
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| * The two bimedians are perpendicular if and only if the two diagonals have equal length
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| ==Trigonometric identities==
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| 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>
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| :<math>\sin{A}+\sin{B}+\sin{C}+\sin{D}=4\sin{\frac{A+B}{2}}\sin{\frac{A+C}{2}}\sin{\frac{A+D}{2}}</math>
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| | |
| and
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| :<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>
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| | |
| Also,<ref>''MathPro Press'', "Original Problems Proposed by Stanley Rabinowitz 1963–2005", p. 23, [http://www.mathpropress.com/archive/RabinowitzProblems1963-2005.pdf]</ref>
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| :<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>
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| In the last two formulas, no angle is allowed to be a [[right angle]], since then the [[tangent function]]s are not defined.
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| ==Maximum and minimum properties==
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| 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}}
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| :<math>K\le \tfrac{1}{16}L^2</math>
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| 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.
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| The quadrilateral with given side lengths that has the [[Maxima and minima|maximum]] area is the [[cyclic quadrilateral]].<ref name=Peter/>
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| 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
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| :<math>K=\tfrac{1}{2}pq\sin{\theta}\le \tfrac{1}{2}pq,</math>
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| | |
| where ''θ'' is the angle between the diagonals ''p'' and ''q''. Equality holds if and only if ''θ'' = 90°.
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| If ''P'' is an interior point in a convex quadrilateral ''ABCD'', then
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| :<math>AP+BP+CP+DP\ge AC+BD.</math>
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| 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>Alsina, Claudi and Nelsen, Roger, ''Charming Proofs. A Journey Into Elegant Mathematics'', Mathematical Association of America, 2010, pp. 114, 119, 120, 261.</ref>{{rp|p.120}}
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| ==Remarkable points and lines in a convex quadrilateral==
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| 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>King, James, ''Two Centers of Mass of a Quadrilateral'', [http://www.math.washington.edu/~king/java/gsp/center-mass-quad.html], Accessed 2012-04-15.</ref>
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| 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.
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| 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}}.</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 ''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/>
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| 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/>
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| Another remarkable line in a convex quadrilateral is the [[Newton line]].
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| ==Other properties of convex quadrilaterals==
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| *Let exterior squares be drawn on all sides of a quadrilateral. 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]].
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| *The internal [[Bisection#Angle bisector|angle bisectors]] of a convex quadrilateral either form a [[cyclic quadrilateral]]<ref name=Altshiller-Court/>{{rp|p.127}} or they are [[Concurrent lines|concurrent]]. In the latter case the quadrilateral is a [[tangential quadrilateral]].
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| *For any simple quadrilateral with given edge lengths, there is a [[cyclic quadrilateral]] with the same edge lengths.<ref name=Peter>Thomas Peter, "Maximizing the Area of a Quadrilateral", ''The College Mathematics Journal'', Vol. 34, No. 4 (September 2003), pp. 315–316.</ref>
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| *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>Martin Josefsson, [http://forumgeom.fau.edu/FG2013volume13/FG201305.pdf "Characterizations of Trapezoids"], ''Forum Geometricorum'' 13 (2013) 23–35.</ref>
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| ==Taxonomy==
| |
| A [[Taxonomy (general)|taxonomy]] of quadrilaterals is illustrated by the following graph. Lower forms are special cases of higher forms. Note that "trapezium" here is referring to the British definition (the North American equivalent is a trapezoid), and "kite" excludes the ''concave kite'' (''arrowhead'' or ''dart''). Inclusive definitions are used throughout.
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| :[[File:Quadrilateral hierarchy.png|Taxonomy of quadrilaterals. Lower forms are special cases of higher forms.]]
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| ==See also==
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| *[[Complete quadrangle]]
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| *[[Perpendicular bisector construction]]
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| *[[Saccheri quadrilateral]]
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| ==References==
| |
| {{reflist|2}}
| |
| | |
| ==External links==
| |
| {{Commons category|Tetragons}}
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| *{{MathWorld |urlname=Quadrilateral |title=Quadrilateral}}
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| *[http://www.chrisvantienhoven.nl/mathematics/encyclopedia.html Encyclopedia of Quadri-Figures] by Chris Van Tienhoven
| |
| *[http://www.vias.org/comp_geometry/geom_quad_general.html Compendium Geometry] Analytic Geometry of Quadrilaterals
| |
| *[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]]
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| *[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-web.html A (dynamic) Hierarchical Quadrilateral Tree] at [http://dynamicmathematicslearning.com/JavaGSPLinks.htm Dynamic Geometry Sketches]
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| *[http://mysite.mweb.co.za/residents/profmd/quadclassify.pdf An extended classification of quadrilaterals] at [http://mysite.mweb.co.za/residents/profmd/homepage4.html Dynamic Math Learning Homepage]
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| *[http://comic.socksandpuppets.com/view.php?date=2008-02-08 Quadrilateral Venn Diagram] Quadrilaterals expressed in the form of a Venn diagram, where the areas are also the shape of the quadrilateral they describe.
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| *[http://mzone.mweb.co.za/residents/profmd/classify.pdf The role and function of a hierarchical classification of quadrilaterals] by Michael de Villiers
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| {{Polygons}}
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| [[Category:Quadrilaterals]]
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| [[Category:Polygons]]
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