The Island of the Incenter

Clark Kimberling has greated a great resource for triangle geometers in The Encyclopedia of Triangle Centers, a collection of thousands of triangle centers. One thing this resource is used for is checking known facts about a particular triangle point. I am trying to do something different. In a collection of so many points, there might be collective patterns hitherto missed. All we have to to is figure out how to look. I call this my "mining the Kimberling points" project, the name coming from an often repeated remark of John Conway that "the ore has been mined but the gold is yet to be extracted." One of the results of mining the Kimberling points is what I call the "island of the incenter."

There is a tendency of triangle points to cluster along certain self-isotomic paths, for example, the self isotomic conic y² – zx = 0. A feature of this curve is that if P = (l:m:n) is on this curve, so are Pⁱ = (lⁱ:mⁱ:nⁱ) where i is any integer. This can put a large number of centers related to (l:m:n) on this curve. I call this grouping of points "the orbit of P." But the orbit is only the beginning.

If another point is near the orbit, then all its powers will closely follow the orbit as will many of the barycentric products of these points.

What is shown in this picture is the product of orbit of the inenter with that of the Nagel point, all of which will have coordinates of the form : bⁿs_b^m :, and products with the orthocenter, which wll have coordinates of the form : bⁿS_B^m :. The Kimberling X numbers are placed next to many of these points.

Note: this would not have been possible without Edward Brisse's translation of the X points into barycentrics. While I am thanking people, I should thank my former students David Akers and Michael Matthews with whom ten years ago I first saw the orbit patterns but was too much of a rookie to realize what we were seeing.

This "island" really interests me because it seems to me to come from a different place than most of the relationships we discuss in triangle geometry.