Excircles and Excenters

A visual introduction to the three excircles and excenters of a triangle, showing how external angle bisectors determine these special circles.

Keywords: excircle, excenter, external angle bisector, tangent lengths

Prerequisites: incenter-proof, angle-bisect-proof · Difficulty: intermediate

Every triangle has not just one inscribed circle (the incircle), but also three excircles — circles tangent to one side and the extensions of the other two sides.

Excenters

Each excircle has a center called an excenter:

• $I_A$: excenter opposite vertex $A$ — its excircle is tangent to $BC$
• $I_B$: excenter opposite vertex $B$ — its excircle is tangent to $CA$
• $I_C$: excenter opposite vertex $C$ — its excircle is tangent to $AB$

Each excenter lies at the intersection of one internal angle bisector and two external angle bisectors.

Tangent Lengths

The tangent length from a vertex to a circle is the distance from the vertex to where a tangent from it touches the circle — not to be confused with the radius. For the excircles, these tangent lengths have clean expressions in terms of the semi-perimeter $s = (a+b+c)/2$.

Review the Incircle

We begin by recalling the incircle. The internal angle bisectors of all three angles meet at the incenter $I$. The incircle is tangent to all three sides from the inside.

External Angle Bisectors

Each angle also has an external bisector — it bisects the exterior angle formed by one side and the extension of the other.

The excenter opposite vertex $A$ lies at the intersection of the internal bisector at $A$ with the external bisectors at $B$ and $C$.

Excircle Opposite $A$

The excircle opposite $A$ is centered at $I_A$, the intersection of the internal bisector at $A$ with the external bisectors at $B$ and $C$.

This circle is tangent to side $BC$ and the extensions of sides $AB$ and $CA$.

All Three Excircles

Applying the same construction for the other two vertices gives two more excircles. Every triangle has exactly three excircles, one opposite each vertex.

Tangent Lengths

The tangent length from a vertex to a circle is the distance along a tangent line from that vertex to the point of tangency (distinct from the radius of the circle).

For the excircle opposite $A$ (centered at $I_A$), the tangent lengths from each vertex are:

• From $A$: tangent length $= s$
• From $B$: tangent length $= s - c$
• From $C$: tangent length $= s - b$

where $a = BC$, $b = CA$, $c = AB$ and $s = (a+b+c)/2$ is the semi-perimeter.

These tangent lengths are fundamental in proving results like the Nagel point and Feuerbach's theorem.

Conclusion

Each triangle has three excircles, tangent to one side and the extensions of the other two. The excenters lie at intersections of internal and external angle bisectors.

The incircle and excircles are deeply connected — Feuerbach's theorem states that the nine-point circle is tangent to all four of these circles!