Steiner point (triangle) explained

In triangle geometry, the Steiner point is a particular point associated with a triangle.^[1] It is a triangle center^[2] and it is designated as the center X(99) in Clark Kimberling's Encyclopedia of Triangle Centers. Jakob Steiner (1796–1863), Swiss mathematician, described this point in 1826. The point was given Steiner's name by Joseph Neuberg in 1886.^[2] ^[3]

Definition

The Steiner point is defined as follows. (This is not the way in which Steiner defined it.^[2])

Let be any given triangle. Let be the circumcenter and be the symmedian point of triangle . The circle with as diameter is the Brocard circle of triangle . The line through perpendicular to the line intersects the Brocard circle at another point . The line through perpendicular to the line intersects the Brocard circle at another point . The line through perpendicular to the line intersects the Brocard circle at another point . (The triangle is the Brocard triangle of triangle .) Let be the line through parallel to the line, be the line through parallel to the line and be the line through parallel to the line . Then the three lines, and are concurrent. The point of concurrency is the Steiner point of triangle .

In the Encyclopedia of Triangle Centers the Steiner point is defined as follows;

Let be any given triangle. Let be the circumcenter and be the symmedian point of triangle . Let be the reflection of the line in the line, be the reflection of the line in the line and be the reflection of the line in the line . Let the lines and intersect at, the lines and intersect at and the lines and intersect at . Then the lines, and are concurrent. The point of concurrency is the Steiner point of triangle .

Trilinear coordinates

The trilinear coordinates of the Steiner point are given below.

Properties

The Steiner circumellipse of triangle, also called the Steiner ellipse, is the ellipse of least area that passes through the vertices, and . The Steiner point of triangle lies on the Steiner circumellipse of triangle .
Canadian mathematician Ross Honsberger stated the following as a property of Steiner point: The Steiner point of a triangle is the center of mass of the system obtained by suspending at each vertex a mass equal to the magnitude of the exterior angle at that vertex.^[4] The center of mass of such a system is in fact not the Steiner point, but the Steiner curvature centroid, which has the trilinear coordinates

\left(	\pi-A
	a

	\pi-B
	b

	\pi-C
	c

\right)

.^[5] It is the triangle center designated as X(1115) in Encyclopedia of Triangle Centers.

The Simson line of the Steiner point of a triangle is parallel to the line where is the circumcenter and is the symmmedian point of triangle .

Tarry point

The Tarry point of a triangle is closely related to the Steiner point of the triangle. Let be any given triangle. The point on the circumcircle of triangle diametrically opposite to the Steiner point of triangle is called the Tarry point of triangle . The Tarry point is a triangle center and it is designated as the center X(98) in Encyclopedia of Triangle Centers. The trilinear coordinates of the Tarry point are given below:

where is the Brocard angle of triangle

and

Similar to the definition of the Steiner point, the Tarry point can be defined as follows:

Let be any given triangle. Let be the Brocard triangle of triangle . Let be the line through perpendicular to the line, be the line through perpendicular to the line and be the line through perpendicular to the line . Then the three lines, and are concurrent. The point of concurrency is the Tarry point of triangle .

Notes and References

Web site: Paul E. Black. Steiner point. Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology.. 17 May 2012.
Web site: Kimberling. Clark. Steiner point. 17 May 2012.
J. Neuberg. Sur le point de Steiner. Journal de mathématiques spéciales. 1886. 29.
Book: Honsberger, Ross. Episodes in nineteenth and twentieth century Euclidean geometry. 1965. The Mathematical Association of America. 119–124.
Web site: Eric W.. Weisstein. Steiner Curvature Centroid. MathWorld—A Wolfram Web Resource.. 17 May 2012.