The Gergonne points, the Nagel points, and the triangle ABC combine to form an interrelated set of points known as a desmic system, shown below. The term "desmic" arose because the Gergonne and Nagel points come in families of 4. To make this work best a fourth point is added to A, B, and C which John Conway named "the Desmon" from the Greek work for "linking." In this case the desmon gets the letter D and is the isotomic conjugate of the orthocenter.
The points in a desmic stystem (the 16 line-12 point projective configuration) can be arranged as a projective cube, of which this picture is a particularly good example.
| o | a | b | c | |
| 0 | D | A | B | C |
| 1 | No | Na | Nb | Nc |
| 2 | Go | Ga | Gb | Gc |
How this chart shows the colinarities between these points is explained on the page on cubics. In this picture the Gergonne points form four vertices of a cube with ABCD being the opposite vetices. The edges continued concur at three of the Nagel points, which become the perspecive points for the cube. The original Nagel point is the at the concurrance of the diagonals.
