Incircle#
Definition: Incircle
An incircle of a [[Polygons|polygon]] is a [[Circle]] in the polygon's [[Planes|plane]] which is [[Configurations of Lines and Circles|tangent]] to all of its [[Polygons|sides]].
Definition: Inradius
A [[Circle|radius]] of an [[Tangential Polygons|incircle]] is known as an inradius.
Definition: Incenter
The [[Circle|center]] of an [[Tangential Polygons|incircle]] is known as an incenter.
Tangential Polygon#
Definition: Tangential Polygon
A [[Polygons|polygon]] is tangential or circumscribed iff it has an [[Tangential Polygons|incircle]].
Theorem: Uniqueness of the Incircle
The [[Tangential Polygons|incircle]] of a [[Tangential Polygon]] is unique.
Proof
TODO
Theorem
A [[Polygons|polygon]] is [[Tangential Polygons|tangential]] if and only if the [[Angle Bisector of a Plane Angle|bisectors]] of its [[Polygons|interior angles]] are all [[Concurrent Lines|concurrent]].
Proof
TODO
Theorem: Incenter of a Tangential Polygon
The [[Tangential Polygons|incenter]] of a [[Tangential Polygons|tangential polygon]] is the [[Euclidean Geometry|point]] where the [[Angle Bisector of a Plane Angle|bisectors]] of the polygon's [[Polygons|interior angles]] intersect.
Proof
TODO
Properties#
Theorem: Convexity of Tangential Polygons
Every [[Tangential Polygons|tangential polygon]] is [[Convex Polygon|convex]].
Proof
TODO
Theorem: Area of a Tangential Polygon
The [[Polygons|area]] \(S\) of a [[Tangential Polygons|tangential polygon]] can be calculated using its [[Polygons|perimeter]] and [[Tangential Polygons|inradius]] \(r\):
Proof
TODO