Tangential polygon


In Euclidean geometry, a tangential polygon, also known as a circumscribed polygon, is a convex polygon that contains an inscribed circle. 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.

Characterizations

A convex polygon has an incircle if and only if all of its internal angle bisectors are concurrent. This common point is the incenter.
There exists a tangential polygon of n sequential sides a1,..., an if and only if the system of equations
has a solution in positive reals. If such a solution exists, then x1,..., xn are the tangent lengths of the polygon.

Uniqueness and non-uniqueness

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.

Inradius

If the n sides of a tangential polygon are a1,..., an, the inradius is
where K is the area of the polygon and s is the semiperimeter.

Other properties

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.

Tangential quadrilateral

Tangential hexagon