Pappus configuration

Pappus's hexagon theorem states that every two triples of collinear points ABC and abc (none of which lie on the intersection of the two lines) can be completed to form a Pappus configuration, by adding the six lines Ab, aB, Ac, aC, Bc, and bC, and their three intersection points X = Ab · aB, Y = Ac · aC, and Z = Bc · bC.

[2] The Pappus configuration can also be derived from two triangles △XcC and △YbB that are in perspective with each other (the three lines through corresponding pairs of points meet at a single crossing point) in three different ways, together with their three centers of perspectivity Z, a, and A.

It is a bipartite symmetric cubic graph with 18 vertices and 27 edges.

The nine points of the Pappus configuration form only nine three-point lines.

However, they can be arranged so that there is another three-point line, making a total of ten.