The Classical Weak points – friends of the incenter

The heroic period of the 19the century saw the creation of the modern geometry of the triangle. One of its residues was the discovery of new triangle centers, named after the geometers of the period. Many of these points have fundamentally different properties than the 4 centers known to Euclid and were named "weak" points by John Conway. Of these the most significant are the Gergonne and Nagel points, but, if they are the leaders, there are many, many followers.

All the classical weak points are derived from the incenters and incircles. Many of these points lie close to a particular path through the triangle that I call the orbit of the incenter.

This is my attempt to create a picture that shows many of the relationships between these points. The affine parallelogram of the Nagel point N_o is used as a device to organize this picture of the most famous of the weak points. The Mineur conic of this parallelogram is the Feuerbach hyperbola, which has F_o as its center. See the bottom of the page linked here for the notation we use. See here for the weak Triangles of Centers, a related organization of the weak points.

Many of these centers are on the affine parallelogram and the triangles, all with centroid G, defined from it. Since weak points are often defined in connection with strong points, some strong points are shown. Those on the Euler line are noted in larger circles.