Editing Kite (geometry) (section)

== Properties ==
=== Diagonals, angles, and area ===
Every kite is an [[orthodiagonal quadrilateral]], 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.{{r|halsted}} Because of its symmetry, the other two angles of the kite must be equal.{{r|beamer|alexander-koeberlein}} The diagonal symmetry axis of a convex kite divides it into two [[congruent triangles]]; the other diagonal divides it into two [[isosceles triangle]]s.{{r|halsted}}

As is true more generally for any orthodiagonal quadrilateral, the area {{tmath|A}} of a kite may be calculated as half the product of the lengths of the diagonals {{tmath|p}} and {{tmath|q}}:{{r|beamer}}
<math display=block>A =\frac{p \cdot q}{2}.</math>
Alternatively, the area can be calculated by dividing the kite into two congruent triangles and applying the [[Area of a triangle#Using trigonometry|SAS formula]] for their area. If <math>a</math> and <math>b</math> are the lengths of two sides of the kite, and <math>\theta</math> is the [[angle]] between, then the area is{{r|crux}}
<math display=block>\displaystyle A = ab \cdot \sin\theta.</math>

=== Inscribed circle ===
{{multiple_image|direction=vertical
|image1=Kite inexcircles.svg
|image2=Dart inexcircles.svg
|image3=Antipar inexcircles.svg
|width=300px
|footer=Two circles tangent to the sides and extended sides of a convex kite (top), non-convex kite (middle), and [[antiparallelogram]] (bottom). The four lines through the sides of each quadrilateral are [[bitangent]]s of the circles.}}
Every ''convex'' kite is also a [[tangential quadrilateral]], a quadrilateral that has an [[inscribed circle]]. That is, there exists a circle that is [[tangent]] to all four sides. Additionally, if a convex kite is not a rhombus, there is a circle outside the kite that is tangent to the extensions of the four sides; therefore, every convex kite that is not a rhombus is an [[ex-tangential quadrilateral]]. The convex kites that are not rhombi are exactly the quadrilaterals that are both tangential and ex-tangential.{{r|alsina-nelson}} For every ''concave'' kite there exist two circles tangent to two of the sides and the extensions of the other two: 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.{{r|wheeler}}

For a convex kite with diagonal lengths {{tmath|p}} and {{tmath|q}} and side lengths {{tmath|a}} and {{tmath|b}}, the radius {{tmath|r}} of the inscribed circle is <math display=block>r=\frac{pq}{2(a+b)},</math>
and the radius {{tmath|\rho}} of the ex-tangential circle is{{r|alsina-nelson}}
<math display=block>\rho=\frac{pq}{2|a-b|}.</math>

A tangential quadrilateral is also a kite [[if and only if]] any one of the following conditions is true:{{r|josefsson-when}}
* The area is one half the product of the [[diagonal]]s.
* The diagonals are [[perpendicular]]. (Thus the kites are exactly the quadrilaterals that are both tangential and [[orthodiagonal quadrilateral|orthodiagonal]].)
* The two line segments connecting opposite points of tangency have equal length.
* The [[tangential quadrilateral#Special line segments|tangent lengths]], distances from a point of tangency to an adjacent vertex of the quadrilateral, are equal at two opposite vertices of the quadrilateral. (At each vertex, there are two adjacent points of tangency, but they are the same distance as each other from the vertex, so each vertex has a single tangent length.)
* The two [[Quadrilateral#Special line segments|bimedians]], line segments connecting midpoints of opposite edges, have equal length.
* The products of opposite side lengths are equal.
* The center of the incircle lies on a line of symmetry that is also a diagonal.

If the diagonals in a tangential quadrilateral {{tmath|ABCD}} intersect at {{tmath|P}}, and the [[Incircle and excircles of a triangle|incircle]]s of triangles {{tmath|ABP}}}, {{tmath|BCP}}, {{tmath|CDP}}, {{tmath|DAP}} have radii {{tmath|r_1}}, {{tmath|r_2}}, {{tmath|r_3}}, and {{tmath|r_4}} respectively, then the quadrilateral is a kite if and only if{{r|josefsson-when}}
<math display=block>r_1+r_3=r_2+r_4.</math>
If the [[Incircle and excircles of a triangle|excircle]]s to the same four triangles opposite the vertex {{tmath|P}} have radii {{tmath|R_1}}, {{tmath|R_2}}, {{tmath|R_3}}, and {{tmath|R_4}} respectively, then the quadrilateral is a kite if and only if{{r|josefsson-when}}
<math display=block>R_1+R_3=R_2+R_4.</math>

=== Duality ===
[[File:Kite isotrap duality.svg|thumb|upright|A kite and its dual isosceles trapezoid]]
Kites and [[isosceles trapezoid]]s are dual to each other, meaning that there is a correspondence between them that reverses the dimension of their parts, taking vertices to sides and sides to vertices. From any kite, the inscribed circle is tangent to its four sides at the four vertices of an isosceles trapezoid. For any isosceles trapezoid, tangent lines to the circumscribing circle at its four vertices form the four sides of a kite. This correspondence can also be seen as an example of [[polar reciprocation]], a general method for corresponding points with lines and vice versa given a fixed circle. Although they do not touch the circle, the four vertices of the kite are reciprocal in this sense to the four sides of the isosceles trapezoid.{{r|robertson}} The features of kites and isosceles trapezoids that correspond to each other under this duality are compared in the table below.{{r|devilliers-adventures}}

{| class=wikitable style="text-align: center;"
|-
! scope="col" | Isosceles trapezoid
! scope="col" | Kite
|-
| Two pairs of equal adjacent angles
| Two pairs of equal adjacent sides
|-
| Two equal opposite sides
| Two equal opposite angles
|-
| Two opposite sides with a shared perpendicular bisector
| Two opposite angles with a shared angle bisector
|-
| An axis of symmetry through two opposite sides
| An axis of symmetry through two opposite angles
|-
| Circumscribed circle through all vertices
| Inscribed circle tangent to all sides
|}

=== Dissection ===
The [[equidissection]] problem concerns the subdivision of polygons into triangles that all have equal areas. In this context, the ''spectrum'' of a polygon is the set of numbers <math>n</math> such that the polygon has an equidissection into <math>n</math> equal-area triangles. Because of its symmetry, the spectrum of a kite contains all even integers. Certain special kites also contain some odd numbers in their spectra.{{r|kasimitis-stein|jepsen-sedberry-hoyer}}

Every triangle can be subdivided into three right kites meeting at the center of its inscribed circle. More generally, a method based on [[circle packing]] can be used to subdivide any polygon with <math>n</math> sides into <math>O(n)</math> kites, meeting edge-to-edge.{{r|bern-eppstein}}