Editing Kite (geometry) (section)

=== 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>