The lines containing the three points of the intersection of the three pairs of opposite
sides of a (not necessarily regular) hexagon.

There are 6! (i.e., 6 factorial) possible ways of taking all polygon vertices in any order, but among these
are six equivalent cyclic permutations and
two possible orderings, so the total number of different hexagons (not all simple)
is

There are therefore a total of 60 Pascal lines created by connecting polygon
vertices in any order.

The 60 Pascal lines form a very complicated pattern which can be visualized most easily in the degenerate case of a regular hexagon inscribed in a circle, as illustrated
above for magnifications ranging over five powers of 2. Only 45 lines are visible
in this figure since each of the three thick lines (located at angles to each other) represents a degenerate group
of four Pascal lines, and six of the Pascal lines are lines
at infinity (Wells 1991).

The pattern for a general ellipse and hexagon (illustrated above) is much more complicated, and is difficult to distinguish from a clutter of lines.

The 60 Pascal lines intersect three at a time through 20 Steiner points (some of which are shown as the
filled circles in the above figures). In the symmetrical case of the regular hexagon
inscribed in a circle, the 20 Steiner points degenerate
into seven distinct points arranged at the vertices and center of a regular hexagon
centered at the origin of the circle. The 60 Pascal line also intersect
three at a time in 60 Kirkman points. Each Steiner
point lines together with three Kirkman points on a total of 20 Cayley
lines. There is a dual relationship between the 60 Pascal lines and the 60 Kirkman points.