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).

There exists a tangential polygon of n sequential sides a₁, ..., a_n if and only if the system of equations

has a solution (x₁, ..., x_n) in positive reals. If such a solution exists, then x₁, ..., 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 number n of sides is odd, then for any given set of sidelengths ${\displaystyle a_{1},\dots ,a_{n}}$ 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.

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