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[[File:Bicentric kite 001.svg|thumb|right|A right kite with its circumcircle and incircle]]
In [[Euclidean geometry]], a '''right kite''' is a [[kite (geometry)|kite]] (a [[quadrilateral]] whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other) that can be inscribed in a circle.<ref name=Villiers>Michael de Villiers, ''Some Adventures in Euclidean Geometry'', ISBN 978-0-557-10295-2, 2009, pp. 154, 206.</ref> That is, it is a kite with a [[Circumscribed circle|circumcircle]] (i.e., a [[cyclic quadrilateral|cyclic]] kite). Thus the right kite is a [[Convex and concave polygons|convex]] quadrilateral and has two opposite [[right angle]]s.<ref>{{citation
|last=De Villiers |first=Michael
|issue=1
|journal=For the learning of mathematics
|jstor=40248098
|pages=11–18
|title=The role and function of a hierarchical classification of quadrilaterals
|volume=14
|year=1994}}</ref> If there are exactly two right angles, each must be between sides of different lengths. All right kites are [[bicentric quadrilateral]]s (quadrilaterals with both a circumcircle and an incircle), since all kites have an [[incircle]]. One of the diagonals (the one that is a line of [[symmetry]]) divides the right kite into two [[right triangle]]s and is also a [[diameter]] of the circumcircle.

In a [[tangential quadrilateral]] (one with an incircle), the four line segments between the center of the incircle and the points where it is tangent to the quadrilateral partition the quadrilateral into four right kites.

==Special case==
A special case of right kites are [[square (geometry)|square]]s, where the diagonals have equal lengths, and the incircle and circumcircle are [[concentric]].

==Characterizations==
A kite is a right kite [[if and only if]] it has a circumcircle (by definition). This is equivalent to its being a kite with two opposite right angles.

==Metric formulas==
Since a right kite can be divided into two right triangles, the following metric formulas easily follow from well known properties of right triangles. In a right kite ''ABCD'' where the opposite angles ''B'' and ''D'' are right angles, the other two angles can be calculated from
:<math>\tan{\frac{A}{2}}=\frac{b}{a},\qquad \tan{\frac{C}{2}}=\frac{a}{b}</math>

where ''a'' = ''AB'' = ''AD'' and ''b'' = ''BC'' = ''CD''. The [[area]] of a right kite is
:<math>\displaystyle K=ab. </math>

The [[diagonal]] ''AC'' that is a line of symmetry has the length
:<math>p=\sqrt{a^2+b^2}</math>

and, since the diagonals are [[perpendicular]] (so a right kite is an [[orthodiagonal quadrilateral]] with area <math>K=\frac{pq}{2}</math>), the other diagonal ''BD'' has the length
:<math>q=\frac{2ab}{\sqrt{a^2+b^2}}.</math>

The [[radius]] of the circumcircle is (according to the [[Pythagorean theorem]])
:<math>R=\tfrac{1}{2}\sqrt{a^2+b^2}</math>

and, since all kites are [[tangential quadrilateral]]s, the radius of the incircle is given by
:<math>r=\frac{K}{s}=\frac{ab}{a+b}</math>

where ''s'' is the semiperimeter.

The area is given in terms of the circumradius ''R'' and the inradius ''r'' as<ref name=MJFG>{{citation
|last=Josefsson |first=Martin
|journal=Forum Geometricorum
|pages=237–241
|title=Maximal Area of a Bicentric Quadrilateral
|url=http://forumgeom.fau.edu/FG2012volume12/FG201222.pdf
|volume=12
|year=2012}}.</ref>
:<math>K=r(r+\sqrt{4R^2+r^2}).</math>

==Duality==
The [[dual polygon]] to a right kite is an [[Tangential trapezoid#Isosceles tangential trapezoid|isosceles tangential trapezoid]].<ref name=Villiers/>

==Alternative definition==
Sometimes a right kite is defined as a kite with at least one right angle.<ref>1728 Software Systems, ''Kite Calculator'', accessed 8 October 2012, [http://www.1728.org/quadkite.htm]</ref> If there is only one right angle, it must be between two sides of equal length; in this case, the formulas given above do not apply.

==References==
{{reflist}}

[[Category:Quadrilaterals]]