Patterns in the X points
Kimberling's X points can be used in several ways. The most common way is to find information about particular points. But there are so many of them and we can think about whole groups of them. I think I have a way to relate points that are not usually thought of together.
Consider this first picture of three which shows the pattern of points not usually considered related. I call these paths "orbits." The numbers next to the points are Kimberling's X numbers. The point is the incredible regularity of the pattern.
Kimberling compiled the points with lots of information, but it was Edward Brisse's organization of those points as barycentrics that made it possible to see the patterns that made this chart possible
There are three pictures in this series for which the link is at the bottom of the page.
These points arise from considering affine relationships between the Gergonne, Nagel, and incenter points.