Every triangle has exactly one circle that touches all three sides: its incircle. The green lines are the angle bisectors. They always meet in a single point, the incenter (gray) — the center of the incircle. A perpendicular from the incenter to a side gives the green point where the circle touches.
The source code is at github.com/KirillOsenkov/LiveGeometry. It is written in .NET and Avalonia.