Convex polygon that contains an inscribed circle
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.
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" (the center of the incircle).
There exists a tangential polygon of "n" sequential sides "a"1, ..., "a""n" if and only if the system of equations
formula_1
has a solution ("x"1, ..., "x""n") in positive reals. 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).
Uniqueness and non-uniqueness.
If the number of sides "n" is odd, then for any given set of sidelengths formula_2 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 "a"1, ..., "a""n", the inradius (radius of the incircle) is
formula_3
where "K" is the area of the polygon and "s" is the semiperimeter. (Since all triangles are tangential, this formula applies to all triangles.)
Tangential triangle.
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.
References.
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