The group table for the Neuberg cubic

Bernard's page on the Neuberg cubic was a revelation for me. I knew that there were many more known points on this cubic, but I was unprepared for what I found on his site, especially the newer stuff. What a wonder-full page.

For me all the action in cubics is in the group table, so here it is: the 112 point cubic. On the right of the table are comments about the construction of the points. I have left out the triangles for many new point in lieu of these notations. (1, 2: Io, g∞) means the triangle is found from the intersections determined by the rows indexed 1 and 2 whose lead members are Io and g∞. These triangles are indexed laboriously on his site, but of course the group table does this job effortlessly. The notations to the left of the table have to do with the name of the point. There are 112 points represented in this table. More comments below.

Note that I have added a few points to this table. X1276 and X1277, which Kimberling calls the 1st and 2nd Evan's perspectors, have simple indices in the group law. These are the perspectors of the reflection of ABC in the triangle edges with the outer and inner vertices of the equilateral triangles erected on the edges. I think of them as the interaction of the incenters with the Isodynamic points. Well the same should happen for the interaction of the incenters with the Fermat points, so I have added them to the table with indices n and s. There are more obvious points to add, such as the tangentials of the Fermats and Isodynamics, which would have indices of ±2s and ±2n.

A comment on the Fermat and Isodynamic points. These points have an extra square root in their definition, namely √3 Area. It is important to realize that the area carries its own square roots and should be included. These square roots makes it difficult to combine well with points that do not have these roots, so these points tend to form their own cycles among them selves and not to interact with the points that do not have these roots. This leads to the variables n and s in the group table, which restrict the possible interactions among points. Compare this group table to that of the Darboux cubic.