1. A circle passes through the incenters of all the triangles.
2. A circle passes through the centroids of all the triangles.
3. If you draw the incircles inside all of the triangles a circle can be drawn tangent to the outside of all of them and tangent to the circle passing through the vertices of the polygon.

So my basic questions are

1. Are these relations true or just very close on my CAD system?
2. If true what are the formulas to calculate the radii of these circles?
3. How does one locate the centers of these circles?

I suggest that it would be helpful to express distances in terms of setting the circumradius of the polygon equal to 1.

Easy to state, much harder to solve! I suspect there are some interesting relations to be found here.

https://www.cut-the-knot.org/Curriculum/Geometry/CirclesInRegularPolygon.shtml

Have not thought of this yet. The first impression is that, while 2 and 3 are probably true, 1 may not be, although, of course, it all may be appearnces within the possible error range.

Need to think of that. Thank you for bringing this up.