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).
There exists a tangential polygon of n sequential sides of lengths a<sub>1</sub>, ..., a<sub>n</sub> if and only if the system of equations
has a solution (x<sub>1</sub>, ..., x<sub>n</sub>) in positive reals. If such a solution exists, then x<sub>1</sub>, ..., x<sub>n</sub> 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 number of sides n is odd, then for any given set of sidelengths satisfying the existence criterion above there is only one tangential polygon. But if n is even there are an infinitude of them. For example, in the quadrilateral case where all sides are equal we can have a rhombus with any value of the acute angles, and all rhombi are tangential to an incircle.
If the n sides of a tangential polygon are a<sub>1</sub>, ..., a<sub>n</sub>, the inradius (radius of the incircle) is
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.