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(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.

Definition
[[File:Steiner point construction 01 .svg|thumb|300px|Construction of the Steiner point.

Lines concurring at the Steiner point:

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


 * Let $ABC$ be any given triangle. Let $A'B'C'$ be the circumcenter and $ABC$ be the symmedian point of triangle $ABC$. The circle with $O$ as diameter is the Brocard circle of triangle $ABC$. The line through $LA$ perpendicular to the line $A$ intersects the Brocard circle at another point $B'C'$. The line through $LB$ perpendicular to the line $B$ intersects the Brocard circle at another point $C'A'$. The line through $LC$ perpendicular to the line $C$ intersects the Brocard circle at another point $A'B'$. (The triangle $ABC$ is the Brocard triangle of triangle $O$.) Let $K$ be the line through $ABC$ parallel to the line $OK$, $ABC$ be the line through $O$ parallel to the line $BC$ and $A'$ be the line through $O$ parallel to the line $CA$. Then the three lines $B'$, $O$ and $AB$ are concurrent. The point of concurrency is the Steiner point of triangle $C'$.

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


 * Let $A'B'C'$ be any given triangle. Let $ABC$ be the circumcenter and $LA$ be the symmedian point of triangle $A$. Let $B'C'$ be the reflection of the line $LB$ in the line $B$, $C'A'$ be the reflection of the line $LC$ in the line $C$ and $A'B'$ be the reflection of the line $LA$ in the line $LB$. Let the lines $LC$ and $ABC$ intersect at $ABC$, the lines $O$ and $K$ intersect at $ABC$ and the lines $lA$ and $OK$ intersect at $BC$. Then the lines $lB$, $OK$ and $CA$ are concurrent. The point of concurrency is the Steiner point of triangle $lC$.

Trilinear coordinates
The trilinear coordinates of the Steiner point are given below.



Properties

 * 1) The Steiner circumellipse of triangle $OK$, also called the Steiner ellipse, is the ellipse of least area that passes through the vertices $AB$, $lB$ and $lC$. The Steiner point of triangle $A″$ lies on the Steiner circumellipse of triangle $lC$.
 * 2) 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 $$\left(\frac{\pi - A}{a} : \frac{\pi - B}{b} : \frac{\pi - C}{c}\right)$$. It is the triangle center designated as X(1115) in Encyclopedia of Triangle Centers.
 * 3) The Simson line of the Steiner point of a triangle $lA$ is parallel to the line $B″$ where $lA$ is the circumcenter and $lB$ is the symmmedian point of triangle $C″$.

Tarry point


The Tarry point of a triangle is closely related to the Steiner point of the triangle. Let $AA″$ be any given triangle. The point on the circumcircle of triangle $BB″$ diametrically opposite to the Steiner point of triangle $CC″$ 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:


 * where $bc / (b^2 - c^2) : ca / (c^2 - a^2) : ab / (a^2 - b^2)$ is the Brocard angle of triangle $= b^2 c^2 \csc(b - C) : c^2 a^2 \csc(c - a) : a^2 b^2 \csc(a - b)$
 * and $ABC$
 * and $A$

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


 * Let $B$ be any given triangle. Let $C$ be the Brocard triangle of triangle $ABC$. Let $ABC$ be the line through $ABC$ perpendicular  to the line $OK$, $O$ be the line through $K$ perpendicular  to the line $ABC$ and $A$ be the line through $B'C'$ perpendicular  to the line $B$. Then the three lines $C'A'$, $C$ and $A'B'$ are concurrent. The point of concurrency is the Tarry point of triangle $ABC$.