Tangential Polygons

Incircle

Definition: Incircle

An incircle of a polygon is a circle in the polygon’s plane which is tangent to all of its sides.

Definition: Inradius

A radius of an incircle is known as an inradius.

Definition: Incenter

The center of an incircle is known as an incenter.

Tangential Polygon

Definition: Tangential Polygon

A polygon is tangential or circumscribed iff it has an incircle.

Theorem: Uniqueness of the Incircle

The incircle of a tangential polygon is unique.

PROOF

TODO

Theorem

A polygon is tangential if and only if the bisectors of its interior angles are all concurrent.

PROOF

TODO

Theorem: Incenter of a Tangential Polygon

The incenter of a tangential polygon is the point where the bisectors of the polygon’s interior angles intersect.

PROOF

TODO

Properties

Theorem: Convexity of Tangential Polygons

Every tangential polygon is convex.

PROOF

TODO

Theorem: Area of a Tangential Polygon

The area $S$ of a tangential polygon can be calculated using its perimeter and inradius $r$:

$S=\frac{P\cdot r}{2}$

PROOF

TODO