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| {{Infobox polygon
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| | name = Kite
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| | image = GeometricKite.svg
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| | caption = A kite showing its sides equal in length and its inscribed circle.
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| | type = [[Quadrilateral]]
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| | euler =
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| | edges = 4
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| | schläfli =
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| | wythoff =
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| | coxeter =
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| | symmetry = [[Reflection symmetry|''D''<sub>''1''</sub>]] (*)
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| | area =
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| | angle =
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| | dual = [[Isosceles trapezoid]]
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| | properties = }}
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| In [[Euclidean geometry]], a '''kite ''' is a [[quadrilateral]] whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other. In contrast, a [[parallelogram]] also has two pairs of equal-length sides, but they are opposite each other rather than adjacent. Kite quadrilaterals are named for the wind-blown, flying [[kite]]s, which often have this shape and which are in turn named for a [[kite (bird)|bird]]. Kites are also known as '''deltoids''', but the word "deltoid" may also refer to a [[deltoid curve]], an unrelated geometric object.
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| A kite, as defined above, may be either [[Convex and concave polygons|convex or concave]], but the word "kite" is often restricted to the convex variety. A concave kite is sometimes called a "dart" or "arrowhead", and is a type of [[pseudotriangle]].
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| ==Special cases==
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| [[File:Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg|thumb|The [[deltoidal trihexagonal tiling]] is made of identical kite faces, with 60-90-120 degree internal angles.]]
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| If all four sides of a kite have the same length (that is, if the kite is [[equilateral]]), it must be a [[rhombus]].
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| If a kite is [[equiangular polygon|equiangular]], meaning that all four of its angles are equal, then it must also be equilateral and thus a [[square (geometry)|square]].
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| A kite with three equal 108° angles and one 36° angle forms the [[convex hull]] of the [[lute of Pythagoras]].<ref>{{citation|title=The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes|first=David|last=Darling|publisher=John Wiley & Sons|year=2004|isbn=9780471667001|page=260|url=http://books.google.com/books?id=HrOxRdtYYaMC&pg=PA260}}.</ref>
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| The kites that are also [[cyclic quadrilateral]]s (i.e. the kites that can be inscribed in a circle) are exactly the ones formed from two congruent [[right triangle]]s. That is, for these kites the two equal angles on opposite sides of the symmetry axis are each 90 degrees.<ref>{{citation |first=P. |last=Gant |title=A note on quadrilaterals |journal=Mathematical Gazette |volume=28 |issue=278 |pages=29–30 |year=1944 |doi=10.2307/3607362 |publisher=The Mathematical Association |jstor=3607362}}.</ref> These shapes are called [[right kite]]s<ref>{{citation
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| |last=De Villiers |first=Michael
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| |issue=1
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| |journal=For the learning of mathematics
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| |jstor=40248098
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| |pages=11–18
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| |title=The role and function of a hierarchical classification of quadrilaterals
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| |volume=14
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| |year=1994}}</ref> and they are in fact [[bicentric quadrilateral]]s (below to the left).
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| There are only eight polygons that can tile the plane in such a way that reflecting any tile across any one of its edges produces another tile; one of them is a right kite, with 60°, 90°, and 120° angles. The tiling that it produces by its reflections is the [[deltoidal trihexagonal tiling]].<ref>{{citation
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| | last1 = Kirby | first1 = Matthew
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| | last2 = Umble | first2 = Ronald
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| | arxiv = 0908.3257
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| | doi = 10.4169/math.mag.84.4.283
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| | issue = 4
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| | journal = Mathematics Magazine
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| | mr = 2843659
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| | pages = 283–289
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| | title = Edge tessellations and stamp folding puzzles
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| | volume = 84
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| | year = 2011}}.</ref>
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| {|class="wikitable"
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| |[[Image:Bicentric kite 001.svg|250px]]<br>A right kite
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| |[[Image:Reuleaux kite.svg|250px|center]]<br>An equidiagonal kite inscribed in a [[Reuleaux triangle]]
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| |}
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| Among all quadrilaterals, the shape that has the greatest ratio of its [[perimeter]] to its [[diameter]] is an [[equidiagonal quadrilateral|equidiagonal]] kite with angles π/3, 5π/12, 5π/6, 5π/12 (above to the right).<ref>{{citation |first=D.G. |last=Ball |title=A generalisation of π |journal=Mathematical Gazette |volume=57 |issue=402 |year=1973 |pages=298–303 |doi=10.2307/3616052}}; {{citation |first1=David |last1= Griffiths |first2=David |last2=Culpin |title=Pi-optimal polygons |journal=Mathematical Gazette |volume=59 |issue=409 |year=1975 |pages=165–175 |doi=10.2307/3617699}}.</ref>
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| In [[non-Euclidean geometry]], a [[Lambert quadrilateral]] is a right kite with three right angles.<ref>{{citation|title=College Geometry|first=Howard Whitley|last=Eves|publisher=Jones & Bartlett Learning|year=1995|isbn= 9780867204759|page=245|url=http://books.google.com/books?id=B81gnTjNazMC&pg=PA245}}.</ref>
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| ==Characterizations==
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| A [[quadrilateral]] is a kite [[if and only if]] any one of the following conditions is true:
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| *Two disjoint pairs of adjacent sides are equal (by definition).
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| *One diagonal is the perpendicular bisector of the other diagonal.<ref name=Usiskin>Zalman Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition", Information Age Publishing, 2008, pp. 49-52.</ref> (In the concave case it is the extension of one of the diagonals.)
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| *One diagonal is a line of symmetry (it divides the quadrilateral into two congruent triangles).<ref name=Villiers>Michael de Villiers, ''Some Adventures in Euclidean Geometry'', ISBN 978-0-557-10295-2, 2009, pp. 16, 55.</ref>
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| *One diagonal bisects a pair of opposite angles.<ref name=Villiers/>
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| ==Symmetry==
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| The kites are the quadrilaterals that have an [[Reflection symmetry|axis of symmetry]] along one of their [[diagonal]]s.<ref name="esg">{{citation |title=Elementary Synthetic Geometry |first=George Bruce |last=Halsted |publisher=J. Wiley & sons |year=1896 |contribution=Chapter XIV. Symmetrical Quadrilaterals |url=http://books.google.com/books?id=H3ALAAAAYAAJ&pg=PA49 |pages=49–53 |authorlink=G. B. Halsted}}.</ref> Any [[simple polygon|non-self-crossing]] quadrilateral that has an axis of symmetry must be either a kite (if the axis of symmetry is a diagonal) or an [[isosceles trapezoid]] (if the axis of symmetry passes through the midpoints of two sides); these include as special cases the [[rhombus]] and the [[rectangle]] respectively, which have two axes of symmetry each, and the [[Square (geometry)|square]] which is both a kite and an isosceles trapezoid and has four axes of symmetry.<ref name="esg"/> If crossings are allowed, the list of quadrilaterals with axes of symmetry must be expanded to also include the [[antiparallelogram]]s.
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| ==Basic properties==
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| Every kite is [[Orthodiagonal quadrilateral|orthodiagonal]], meaning that its two diagonals are [[perpendicular|at right angles]] to each other. Moreover, one of the two diagonals (the symmetry axis) is the [[perpendicular bisector]] of the other, and is also the [[angle bisector]] of the two angles it meets.<ref name="esg"/>
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| One of the two diagonals of a convex kite divides it into two [[isosceles triangle]]s; the other (the axis of symmetry) divides the kite into two [[congruent triangles]].<ref name="esg"/> The two interior angles of a kite that are on opposite sides of the symmetry axis are equal.
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| ==Area==
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| As is true more generally for any [[orthodiagonal quadrilateral]], the area ''A'' of a kite may be calculated as half the product of the lengths of the diagonals ''p'' and ''q'':
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| :<math>A =\frac{p \cdot q}{2}.</math>
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| Alternatively, if ''a'' and ''b'' are the lengths of two unequal sides, and ''θ'' is the [[angle]] between unequal sides, then the area is
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| :<math>\displaystyle A = ab \cdot \sin{ \theta}.</math>
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| ==Tangent circles==
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| Every ''convex'' kite has an [[inscribed circle]]; that is, there exists a circle that is [[tangent]] to all four sides. Therefore, every convex kite is a [[tangential quadrilateral]]. Additionally, if a convex kite is not a rhombus, there is another circle, outside the kite, tangent to the lines that pass through its four sides; therefore, every convex kite that is not a rhombus is an [[ex-tangential quadrilateral]].
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| For every ''concave'' kite there exist two circles tangent to all four (possibly extended) sides: one is interior to the kite and touches the two sides opposite from the concave angle, while the other circle is exterior to the kite and touches the kite on the two edges incident to the concave angle.<ref>{{citation |first=Roger F. |last=Wheeler |title=Quadrilaterals |journal=Mathematical Gazette |volume=42 |issue=342 |pages=275–276 |year=1958 |doi=10.2307/3610439 |publisher=The Mathematical Association |jstor=3610439}}.</ref>
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| ==Dual properties==
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| Kites and isosceles trapezoids are dual: the [[Dual polyhedron#Polar reciprocation|polar figure]] of a kite is an isosceles trapezoid, and vice versa.<ref>{{citation |first=S.A. |last= Robertson |title=Classifying triangles and quadrilaterals |journal=Mathematical Gazette |volume=61 |issue=415 |pages=38–49 |year=1977 |doi=10.2307/3617441 |publisher=The Mathematical Association |jstor= 3617441}}.</ref> The side-angle duality of kites and isosceles trapezoids are compared in the table below.<ref name=Villiers/>
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| {| class=wikitable
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| |-
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| ! Isosceles trapezoid
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| ! Kite
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| |-
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| | align=center|Two pairs of equal adjacent angles
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| | align=center|Two pairs of equal adjacent sides
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| |-
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| | align=center|One pair of equal opposite sides
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| | align=center|One pair of equal opposite angles
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| |-
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| | align=center|An axis of symmetry through one pair of opposite sides
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| | align=center|An axis of symmetry through one pair of opposite angles
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| |-
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| | align=center|Circumscribed circle
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| | align=center|Inscribed circle
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| |}
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| ==Tilings and polyhedra==
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| All kites [[tessellation|tile the plane]] by repeated inversion around the midpoints of their edges, as do more generally all quadrilaterals. A kite with angles π/3, π/2, 2π/3, π/2 can also tile the plane by repeated reflection across its edges; the resulting tessellation, the [[deltoidal trihexagonal tiling]], superposes a tessellation of the plane by regular hexagons and isosceles triangles.<ref>See {{mathworld |title=Polykite |urlname=Polykite}}.</ref>
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| The [[deltoidal icositetrahedron]], [[deltoidal hexecontahedron]], and [[trapezohedron]] are [[polyhedra]] with congruent kite-shaped [[facet (mathematics)|facets]]. There are an infinite number of [[uniform tilings in hyperbolic plane|uniform tilings]] of the [[hyperbolic geometry#Models_of_the_hyperbolic_plane|hyperbolic plane]] by kites, the simplest of which is the deltoidal triheptagonal tiling.
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| Kites and darts in which the two isosceles triangles forming the kite have apex angles of 2π/5 and 4π/5 represent one of two sets of essential tiles in the [[Penrose tiling]], an [[aperiodic tiling]] of the plane discovered by mathematical physicist [[Roger Penrose]].
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| Face-transivive self-tesselation of the sphere, Euclidean plane, and hyperbolic plane with kites occurs as uniform duals: {{CDD|node_f1|p|node|q|node_f1}} for [[Coxeter group]] [p,q], with any set of p,q between 3 and infinity, as this table partially shows up to q=6. When p=q, the kites become [[rhombi]].
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| {{Deltoidal table}}
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| ==Conditions for when a tangential quadrilateral is a kite==
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| A [[tangential quadrilateral]] is a kite [[if and only if]] any one of the following conditions is true:<ref name=Josefsson>{{citation
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| |last=Josefsson |first=Martin
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| |journal=Forum Geometricorum
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| |pages=165–174
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| |title=When is a Tangential Quadrilateral a Kite?
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| |url=http://forumgeom.fau.edu/FG2011volume11/FG201117.pdf
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| |volume=11
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| |year=2011}}.</ref>
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| *The area is one half the product of the [[diagonal]]s.
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| *The diagonals are [[perpendicular]]. (Thus the kites are exactly the quadrilaterals that are both tangential and [[orthodiagonal quadrilateral|orthodiagonal]].)
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| *The two line segments connecting opposite points of tangency have equal length.
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| *One pair of opposite [[Quadrilateral#Special line segments|tangent lengths]] have equal length.
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| *The [[Quadrilateral#Special line segments|bimedians]] have equal length.
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| *The products of opposite sides are equal.
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| *The center of the incircle lies on a line of symmetry that is also a diagonal.
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| If the diagonals in a tangential quadrilateral ''ABCD'' intersect at ''P'', and the [[Incircle and excircles of a triangle|incircle]]s in triangles ''ABP'', ''BCP'', ''CDP'', ''DAP'' have radii ''r''<sub>1</sub>, ''r''<sub>2</sub>, ''r''<sub>3</sub>, and ''r''<sub>4</sub> respectively, then the quadrilateral is a kite if and only if<ref name=Josefsson/>
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| :<math>r_1+r_3=r_2+r_4.</math>
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| If the [[Incircle and excircles of a triangle|excircle]]s to the same four triangles opposite the vertex ''P'' have radii ''R''<sub>1</sub>, ''R''<sub>2</sub>, ''R''<sub>3</sub>, and ''R''<sub>4</sub> respectively, then the quadrilateral is a kite if and only if<ref name=Josefsson/>
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| :<math>R_1+R_3=R_2+R_4.</math>
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| ==References==
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| {{reflist|2}}
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| ==External links==
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| {{Commons category|Deltoids}}
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| *{{MathWorld |urlname=Kite |title=Kite}}
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| *[http://www.mathopenref.com/kite.html Kite definition] and [http://www.mathopenref.com/kitearea.html area formulae] with interactive animations at Mathopenref.com
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| [[Category:Elementary shapes]]
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| [[Category:Quadrilaterals]]
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| [[Category:Kites]]
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