Mining the Kimberling points

We have a resource unknown to older geometers, the collections of geometric objects that are being compiled by Kimberling and others. With all this information I figure there are ways to find not just new points (which do not interest me greatly) but whole new stuctures (which do).

Distribution of Centers	There is a favored path through the triangle that I call the sweep of the incenter because of its shape and the collection of these points, the island of the incenter because of its shape. This concept is explained here in the article on the non-Euclidean distribution of points. This path goes through G and approximately the orbit of the incenter. Here is a picture of the distribution of triangle centers created by plotting 2400 of the point from Kimberling's Encyclopaedia which clearly shows the "sweeps" pattern.
Looking for absence of structure	One way to look at the thousands of Kimberling X points is to look negatively, to look for what is not there. Centers of the incentral triangle are both not well represented in ETC and those that are have few listed properties. Here is my look at the incentral triangle and similar topics.
Symmetrizing the coordinates	The next way I am trying to get information out of the collection of points is by converting the coordinates of the points to symmetric form. Edward Brisse has converted all the point coordinates to a uniform notation. His points are also barycentric, which gives the idea of symmetry more meaning. All this makes them a wonderful laboratory. Here are some examples of what I mean by "symmetric form." For example Edward Brisse lists X144 has having its a-coordinate as 3a^2 - 2ab - b^2 - 2ac + 2b*c - c^2 I wrote a routine in Mathematica to symmetrize this to s_bc – a s_a which is much nicer and immediately gives connection between well known points, the Nagel and Mittenpunk in this case. My routine in Mathematica currently does 3rd degree and below. I will add 4th degree soon. It is easy to write routines to do this, but difficult to do it well.
Search for identities	Each collineation of triangle points represents an algebraic identity. Since we have so many colineations listed in ETC, perhaps new relationships can be discovered.
Equivalence Classes of centers:	The following three rows represent different attempts to discover equivalence classes of centers.
Strength	I am looking for ways to divide up the Kimberling point into equivalences. The first way to do this is John's extraversion, which divides the points into equivalence classes based on his concept of strength. Most of the points in ETC are strong points, quartile points, and 2-fold (such as the isodynamics or the foci of the steiner ellipse). The extraversion "operation" is different from point to point, according to its equivalence class. There is nothing in traditional geometry like this (although Lemoine envisioned it -- calling it the "abstract theory of the triangle.") Here is a sketchpad file showing extraversion geometrically. I plan to use Edward Brisse's compilation of triangle points to publish a list of the strength of all the ETC points he lists. This will be simple to do once I get into programming mode (hopefullly soon). I have done this!
the Total Coordinate, conormalized points	For any set of points with three homogeneous coordinates, there is a "fourth coordinate" in the sum of the first three. If the point be a center, then this "total coordinate" is a symmetric function of triangle parameters such as the sidelengths or the angles. Points can be grouped into "conormalized' groups by collecting points together whose total coordinates are the same up to multiples. At first these will be integral multiples, but this can be extended to other fields.
Angle functions; "Galois extensions"	The algebraic nature of the coordinates determines the types of relationships one point can have with others. First page on this: The equivalence class of the Fermat/isodynamic points"