Definition:Excircle of Triangle

Definition

Let $\triangle ABC$ be a triangle whose sides are $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively.

Let sides $b$ and $c$ be produced beyond the vertices $C$ and $B$ respectively.

Let a circle be constructed tangent to both of these extensions to $b$ and $c$ lines and also to $a$.

The circle so constructed is called the excircle of $\triangle ABC$ with respect to $a$.

There are three excircles for every triangle.

The center of an excircle of a triangle is called an excenter of the triangle.

In the above diagram, $I_a$ is the excenter of $\triangle ABC$ with respect to $a$.

A radius of an excircle of a triangle is called an exradius of the triangle.

In the above diagram, $r_a$ is the exradius of $\triangle ABC$ with respect to $a$.

Some (usually older) sources hyphenate: ex-circle.

An excircle is also sometimes referred to as an escribed circle.

The word ecircle can also be seen, mainly in the U.S.