Steiner point (triangle)


In triangle geometry, the Steiner point is a particular point associated with a triangle. It is a triangle center and it is designated as the center X in Clark Kimberling's Encyclopedia of Triangle Centers. Jakob Steiner, Swiss mathematician, described this point in 1826. The point was given Steiner's name by Joseph Neuberg in 1886.

Definition

The Steiner point is defined as follows.
In the Encyclopedia of Triangle Centers the Steiner point is defined as follows:

Trilinear coordinates

The trilinear coordinates of the Steiner point are given below.

Properties

  1. 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.
  2. The Simson line of the Steiner point of triangle is parallel to the Brocard axis, the line where is the circumcenter and is the symmmedian point of triangle.
  3. The Steiner point of the Brocard triangle is the symmedian point of. The Tarry point of the Brocard triangle is the circumcenter of.
  4. The Steiner point of triangle is the Brianchon point of the Kiepert parabola with respect to triangle.
  5. The Steiner point of the medial triangle of is the center of the Kiepert hyperbola of triangle.

Misconception

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. 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. It is the triangle center designated as X in Encyclopedia of Triangle Centers.

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 in Encyclopedia of Triangle Centers. The trilinear coordinates of the Tarry point are given below:
Similar to the definition of the Steiner point, the Tarry point can be defined as follows: