Tangential quadrilateral

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An example of a tangential quadrilateral

Chao's characterization in terms of the radii of circles within each of four triangles

In geometry, a tangential quadrilateral is a convex quadrilateral whose sides all lie tangent to a single circle inscribed within the quadrilateral. This circle is called the incircle. Because tangential quadrilaterals can be drawn surrounding or circumscribing their incircles, they are also sometimes called circumscribed quadrilaterals or circumscribable quadrilaterals. Another name for the same class of quadrilaterals is inscriptable quadrilaterals.

1 Example
2 Characterization
3 Area
4 Other properties
5 References
6 External links

[edit] Example

Examples of tangential quadrilaterals are squares, rhombi, and kites.

If a quadrilateral is both tangential and cyclic, it is called bicentric.

[edit] Characterization

In a tangential quadrilateral, the four angle bisectors meet at the center of the inscribed circle, and conversely a convex quadrilateral in which the four angle bisectors meet at a point must be tangential.

In a tangential quadrilateral, the two pairs of opposite sides add to the same total length, which equals the semiperimeter of the quadrilateral:

$a + c = b + d = \frac{a + b + c + d}{2} = s.$

Conversely a convex quadrilateral in which a + c = b + d must be tangential.

Wu Wei Chao (2000) observed another characterization of tangential quadrilaterals. The two diagonals of any convex quadrilateral partition the quadrilateral into four triangles. Let r₁, r₂, r₃, and r₄ denote the radii of the circles inscribed in each of these triangles; then the quadrilateral is tangential if and only if

$\frac{1}{r_1}+\frac{1}{r_3}=\frac{1}{r_2}+\frac{1}{r_4}.$

[edit] Area

The area of a tangential quadrilateral is

$\displaystyle A = rs,$

where s is the semiperimeter described above and r is the circle's radius (called the inradius).

[edit] Other properties

The line segments between the center of the inscribed circle and the points where it is tangent to the quadrilateral partition the quadrilateral into kites.