Strong Conics Hub

This chart shows the logical organization of strong points lines and conics that builds from K, the symmedian point. It includes every strong conic about which there has been discussion that I could find, save one, the isogonal conjugate of the S—T line.

Each circle represents a point, each box a line, and each oval a conic. The connecing lines represent affine invariant operations: t is the isotomic conjugate, m and d are the medial and dilated (antimedial) operations, and ~ means the dual.

The premises of the information displayed on this page is

1. All connections are affine invariant (making the structure a universal one). Surprisingly it seems that affine connections suffice.

2. There is a special conic type which I call the Mineur conic whose important points organized the conic information. The universal patterna is very simple. Choose a point P (equal to K on this page) and generate the sequence dP, tdP. These two points define the Mineur conic on which the listed point reside. Use these points to generate their circum and inconic. This will easily generate every relevant conic, as judged by the history of conic discussions. Here is the corresponding structure for weak conics, generated by I, the incenter.

The isotomic conjugate takes points to points and lines to conics. The dual operation takes points to lines, lines to points, and changes a circumconic to an inconic (and vice-versa). The dual of a point on a circumconic is a tangent line to an inconic (and vice-versa).

Click on the name of a conic to see pictures and information about it. Not all are implemented yet. Gray text indicates conics that are in the logical system but seem not very important in that there has been little discussion of them by the community of geometers.

This organizational structure is a universal structure, just choose a different starting point. Go here for weak conics.