Excenter of a Triangle
The excenter is the point where an internal angle bisector intersects with the external bisectors of the other two non-adjacent angles of the triangle.
The excenter is the center of a circle that is externally tangent to one side of the triangle and the extensions of the other two sides.
A triangle has three excenters, one for each side.
Thus, every triangle has three excircles, each of which is externally tangent to one side and the extensions of the other two sides.
The radius of each excircle is called the exradius.
Note: In the case of equilateral triangles, all excircles have the same radius.
How to Find the Excenter
You can locate a triangle's excenter using the angle bisectors of the triangle.
Consider a triangle ABC.
Draw the internal bisector of angle γ.
Next, draw the external bisectors of the non-adjacent angles α' and β'.
The bisector of an external angle is also called an external bisector. In both cases, it is the segment that divides the supplementary angle (180° - α and 180° - β) into two equal parts. Thus, you can refer to it as the external bisector of angle α' = 180° - α or the external bisector of angle α. The meaning remains the same.
The intersection of the internal bisector of angle γ with one of the external bisectors of angles α and/or β identifies the excenter Ec relative to angle γ.
Once the excenter is found, you can draw a circle centered at Ec that is tangent externally to the opposite side AB and the extensions of the other two sides AC and BC.
The radius of this circle is called the exradius relative to angle γ.
Note: To summarize, the excenter of a triangle relative to a specific angle can be found at the intersection of the internal bisector of that angle and the external bisectors of the other two angles.
Similarly, the excenters relative to angles α and B can be found at the intersection points of their respective internal bisectors with the external bisectors of the other two angles.
For example, here is the excenter relative to angle α.
Finally, here is the excenter relative to angle β.
And so on.