This is an exercise I did in Blender with geometric nodes, as the circumcenter, incenter and barycenter of the triangle move according to the movement of the vertices. This was something similar to what I did in Pascal and Basic programming in the early 90s, and I tried it in Blender just for fun. But I have to say it was harder in Blender than in old programming, there are so many solutions now that it drives me crazy. Anyway, here is my final result and the Blender file.
Triangle_Incenter-Circumcenter-Barycenter.blend (1.6 MB)