The orbit of the incenter

This is a very interesting picture. It shows the pattern formed by the points of barycentric coordinates : bⁿ : for n an integer. This is the orbit of points in the group of triangle centers generated by the incenter where the group operation is barycentric multiplication. I first saw this 10 years ago as two of my students, David Akers (now in graduate school at Stanford) and Michael Matthews (now teaching school) and I worked through Clark Kimberling's article on Central Points and Central Lines in Mathematics Magazine. It was this article that got me interested in Triangle geometry and in homogeneous coordinates, the like of which I had never seen. Despite its striking nature, I initially discounted it because I had never heard of most of the points on the orbit.

Now that I have seen this path, I see it all the time. I call it the "sweep of the incenter." It marks particularly favored real estate in the plane of the triangle. Since so many of the points that we study are derived from the incenter, this region of the triangle is particularly favored by many of the points we study. Examples are here, here, here, here, here, and here.

This picture shows the points very nicely arranged into their orbit. The coordinates are shown in red and Kimberling numbers for points in listed ETC are in blue. The part of the orbit outside the central region is composed of harmonic assocates of the points inside.

This orbit is the invariant curve of the projective transformation that takes ABCG into ABCIo, in that order. This curve can be seen here.

The path containing the orbit is an equation of the form yⁿ – zx = 0.