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# Steiner Points

There are (at least) three different types of points known as Steiner points.

The point 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

 (1) (2)

It also lies on the Steiner circumellipse. It is Kimberling center . 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).

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.

Brianchon Point, Brocard Triangles, Cayley Lines, Circumcircle, Conic Section, Kiepert Parabola, Kirkman Points, Symmedian Point, Pascal Lines, Pascal's Theorem, Plücker Lines, Salmon Points, Steiner Curvature Centroid, Steiner Set, Steiner's Theorem, Steiner Triple System, Tarry Point

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## References

Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 66 and 329, 1893.Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 77, 1971.Eddy, R. H. and Fritsch, R. "The Conics of Ludwig Kiepert: A Comprehensive Lesson in the Geometry of the Triangle." Math. Mag. 67, 188-205, 1994.Gallatly, W. "The Steiner and Tarry Points." §143 in The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 102, 1913.Honsberger, R. "The Steiner Point and the Tarry Point." §10.5 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 119-124, 1995.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 236-237, 281-282, 1929.Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.Kimberling, C. "Steiner Point." http://faculty.evansville.edu/ck6/tcenters/class/steiner.html.Kimberling, C. "Encyclopedia of Triangle Centers: X(99)=Steiner Point." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X99.Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, 1893.Neuberg, J. "Sur le point de Steiner." J. de math. spéciales, p. 29, 1886.Salmon, G. "Notes: Pascal's Theorem, Art. 267" in A Treatise on Conic Sections, 6th ed. New York: Chelsea, pp. 379-382, 1960.Steiner, J. "Questions proposées. Théorèmes sur l'hexagramum mysticum." Ann. Math. 18, 339-340, 1827-1828.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 172, 1991.

Steiner Points

## Cite this as:

Weisstein, Eric W. "Steiner Points." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SteinerPoints.html