Definition:Excircle of Triangle

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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 $a$.


Excircle-of-Triangle.png


There are three excircles for every triangle.


Excenter

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 $a$.


Exradius

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 $a$.


Also known as

An excircle of a triangle can also be referred to as an excircle to a triangle.


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.


Also see

  • Results about excircles of triangles can be found here.


Sources