Editing Pascal's theorem (section)

==''Hexagrammum Mysticum''==
If six unordered points are given on a conic section, they can be connected into a hexagon in 60 different ways, resulting in 60 different instances of Pascal's theorem and 60 different Pascal lines. This [[projective configuration|configuration]] of 60 lines is called the ''Hexagrammum Mysticum''.<ref>{{harvnb|Young|1930|p=67}} with a reference to Veblen and Young, ''Projective Geometry'', vol. I, p. 138, Ex. 19.</ref><ref>{{harvnb|Conway|Ryba|2012}}</ref>

As [[Thomas Kirkman]] proved in 1849, these 60 lines can be associated with 60 points in such a way that each point is on three lines and each line contains three points. The 60 points formed in this way are now known as the '''Kirkman points'''.<ref>{{harvnb|Biggs|1981}}</ref> The Pascal lines also pass, three at a time, through 20 '''Steiner points'''. There are 20 '''Cayley lines''' which consist of a Steiner point and three Kirkman points. The Steiner points also lie, four at a time, on 15 '''Plücker lines'''. Furthermore, the 20 Cayley lines pass four at a time through 15 points known as the '''Salmon points'''.<ref>{{harvnb|Wells|1991|p=172}}</ref>