The 60 Pascal lines of a hexagon inscribed in a conic intersect three at a time through 20 Steiner points, and also three at a time in 60 points known as Kirkman points. Each Steiner point lies together with three Kirkman points on a total of 20 lines known as Cayley lines. There is a reciprocity relationship between the 60 Kirkman points and the 60 Pascal lines (Hesse, quoted in Salmon 1960), although the relationship is not one of duality in the commonly accepted meaning of that word.

Cayley lines | Pascal lines | Pascal's theorem | Plücker lines | Salmon points | Steiner points