বহিঃকেন্দ্র ও বহির্বৃত্ত

This note explains what excircles and excenters are

Let's say we have a triangle $ABC$. Extend sides $AB$ and $BC$ beyond $A$ and $C$ respectively.

The circle that lies outside triangle $ABC$ and is tangent to sides $AC$, and the extensions of $AB$ and $BC$ is called an excircle of triangle $ABC$.

In every triangle, there are three excircles. Each excircle is distinct from each other. Each excircle is tangent to one of the triangle's sides.

The excenters are the centers of each of the three circles.

Pop quiz!:

Sketch of proof: First we show that $A, I, E_{b}$ are collinear. This is because the lengths of perpendiculars dropped from $E_{b}$ to $AB$ and $AC$ respectively are equal. So we conclude that $E_{b}$ must lie on the angle bisector of angle $CAB$.