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.
of concurrence of the three lines drawn through the
vertices of a triangle parallel
to the corresponding sides of the first Brocard
triangle is called the Steiner point (Honsberger 1995). It lies on the circumcircle
opposite the Tarry point 
and has equivalent triangle
center functions
. The Brianchon point
for the Kiepert parabola is also the Steiner
point (Eddy and Fritsch 1994). The symmedian point 
is the Steiner point of the first Brocard triangle (Honsberger 1995, pp. 120-121).
The Simson line of the Steiner point is parallel
to the line 
,
when 
is the circumcenter and 
is the symmedian point
(Honsberger 1995, p. 121).
