Geometry • Topic 15

Excenters

While a triangle has one inscribed circle, it has three escribed circles (excircles) that are tangent to one side and the extensions of the other two.

Definition

Definition (Excenter)

The excenter opposite vertex

A

, denoted

I_{a}

, is the intersection of:

The internal angle bisector of $A$ .
The external angle bisectors of $B$ and $C$ .

Properties

1. Distances

$I_{a}$ is equidistant from side $a$ and lines $b, c$ . This distance is the exradius $r_{a}$ .
Area formulas: $K = r_{a} (s - a) = r_{b} (s - b) = r_{c} (s - c) = rs$ .

2. Relationship with Incenter

The incenter

I

and the three excenters

I_{a}, I_{b}, I_{c}

form an orthocentric system:

$I$ is the orthocenter of $△ I_{a} I_{b} I_{c}$ .
The triangle $A BC$ is the orthic triangle of $△ I_{a} I_{b} I_{c}$ .

3. Fact 5 Revisited

The midpoint

M

of arc

BC

(containing the circumcircle) is the center of a circle passing through

B, I, C, and I_{a}

Practice Problems

Exercise (Problem 1)

In a triangle with sides 3, 4, 5, find the radii of the three excircles.

Exercise (Problem 2)

Prove that

\frac{1}{r} = \frac{1}{r _{a}} + \frac{1}{r _{b}} + \frac{1}{r _{c}}

Exercise (Problem 3)

Let

I_{a}

be the excenter opposite

A

. Prove that

A I_{a}

passes through the midpoint of the arc

BC

not containing

A