A second "Steiner point," more properly known as the Steiner curvature centroid, is the geometric centroid
of the system obtained by placing a mass equal to the magnitude of the exterior angle
at each vertex (Honsberger 1995, p. 120).

A third sort of Steiner point (Steiner 1827-1828; Lachlan 1893, pp. 115-116) arises if triplets of opposites sides on a conic section
in Pascal's theorem are extended for all permutations
of vertices, 60 Pascal lines
are produced. The 20 points of their three by three intersections are called Steiner
points. Steiner's theorem states that these points
are generated by the hexagons 123456, 143652, and 163254 formed by interchanging
the vertices at positions 2, 4, and 6 (where the numbers denote the order in which
the vertices of the hexagon are taken). The configuration of Pascal
lines for a general hexagon inscribed in a general ellipse are shown above, with
Steiner points shown as filled circles. A blow-up of the region in the upper left
figure is shown below, illustrating the concurrence of three Pascal lines at each
Steiner point.

Each Steiner point lies together with three Kirkman points on a total of 20 lines known as Cayley lines.
The Steiner points also lie four at a time on 15 Plücker
lines (Wells 1991). There is a dual relationship between the 20 Steiner points
and the 20 Cayley lines.