Steiner point (triangle)

1. The Steiner circumellipse of triangle ABC, also called the Steiner ellipse, is the ellipse of least area that passes through the vertices A, B and C. The Steiner point of triangle ABC lies on the Steiner circumellipse of triangle ABC.
2. The Simson line of the Steiner point of triangle ABC is parallel to the Brocard axis, the line OK where O is the circumcenter and K is the symmmedian point of triangle ABC.
3. The Steiner point of the Brocard triangle is the symmedian point of ABC. The Tarry point (see below) of the Brocard triangle is the circumcenter of ABC.^[4]
4. The Steiner point of triangle ABC is the Brianchon point of the Kiepert parabola with respect to triangle ABC.^[5]
5. The Steiner point of the medial triangle of ABC is the center of the Kiepert hyperbola of triangle ABC.^[6]

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 ${\displaystyle \left({\frac {\pi -A}{a}}:{\frac {\pi -B}{b}}:{\frac {\pi -C}{c}}\right)}$.^[7] It is the triangle center designated as X(1115) in Encyclopedia of Triangle Centers.

The Tarry point of a triangle is closely related to the Steiner point of the triangle. Let ABC be any given triangle. The point on the circumcircle of triangle ABC diametrically opposite to the Steiner point of triangle ABC is called the Tarry point of triangle ABC. 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:

⁠${\displaystyle \sec(A+\omega ):\sec(B+\omega ):\sec(C+\omega )=f(a,b,c):f(b,c,a):f(c,a,b)}$⁠

where ω is the Brocard angle of triangle ABC

and ⁠${\displaystyle f(a,b,c)={\frac {bc}{b^{4}+c^{4}-a^{2}b^{2}-a^{2}c^{2}}}}$⁠

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

Let ABC be any given triangle. Let A'B'C' be the Brocard triangle of triangle ABC. Let L_A be the line through A perpendicular to the line B'C', L_B be the line through B perpendicular to the line C'A' and L_C be the line through C perpendicular to the line A'B'. Then the three lines L_A, L_B and L_C are concurrent. The point of concurrency is the Tarry point of triangle ABC.