*Incircle, Excircles, Gergonne, and Nagel
Incircles
The center of the incircle is marked with point I.
This applet also introduces excircles: when the sides of a triangle are extended, the excircles are circles that are tangent to the extended sides of the circle AND the third side of the triangle.
For funsies: How do you think you would construct an excircle? Open up another tab of geogebra.org/geometry and see if your idea works!!!