# Tangential polygon

In Euclidean geometry, a **tangential polygon**, also known as a **circumscribed polygon**, is a convex polygon that contains an inscribed circle (also called an *incircle*). This is a circle that is tangent to each of the polygon's sides. The dual polygon of a tangential polygon is a cyclic polygon, which has a circumscribed circle passing through each of its vertices.

All triangles are tangential, as are all regular polygons with any number of sides. A well-studied group of tangential polygons are the tangential quadrilaterals, which include the rhombi and kites.

A convex polygon has an incircle if and only if all of its internal angle bisectors are concurrent. This common point is the *incenter* (the center of the incircle).^{[1]}

There exists a tangential polygon of *n* sequential sides *a*_{1}, ..., *a*_{n} if and only if the system of equations

has a solution (*x*_{1}, ..., *x*_{n}) in positive reals.^{[2]} If such a solution exists, then *x*_{1}, ..., *x*_{n} are the *tangent lengths* of the polygon (the lengths from the vertices to the points where the incircle is tangent to the sides).

If the *n* sides of a tangential polygon are *a*_{1}, ..., *a*_{n}, the inradius (radius of the incircle) is^{[4]}

where *K* is the area of the polygon and *s* is the semiperimeter. (Since all triangles are tangential, this formula applies to all triangles.)

While all triangles are tangential to some circle, a triangle is called the tangential triangle of a reference triangle if the tangencies of the tangential triangle with the circle are also the vertices of the reference triangle.