Steiner point (triangle)

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Template:Short description 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

Construction of the Steiner point. Template:Legend-line Template:Legend-line Template:Legend-line Template:Legend-line
Lines concurring at the Steiner point: Template:Legend-line Template:Legend-line Template:Legend-line

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

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

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

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

Trilinear coordinates

The trilinear coordinates of the Steiner point are given below.

Template:Tmath
Template:Tmath

Properties

  1. The Steiner circumellipse of triangle Template:Mvar, also called the Steiner ellipse, is the ellipse of least area that passes through the vertices Template:Mvar, Template:Mvar and Template:Mvar. The Steiner point of triangle Template:Mvar lies on the Steiner circumellipse of triangle Template:Mvar.
  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.[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 (π−Aa:π−Bb:π−Cc).[5] 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 Template:Mvar is parallel to the line Template:Mvar where Template:Mvar is the circumcenter and Template:Mvar is the symmmedian point of triangle Template:Mvar.

Tarry point

The line through Template:Mvar perpendicular to Template:Mvar, the line through Template:Mvar perpendicular to Template:Mvar, and the line through Template:Mvar perpendicular to Template:Mvar concur at the Tarry point.

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

Template:Tmath
where Template:Mvar is the Brocard angle of triangle Template:Mvar
and Template:Tmath

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

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

References

Template:Reflist

  1. ↑ Template:Cite web
  2. ↑ 2.0 2.1 2.2 Template:Cite web
  3. ↑ Template:Cite journal
  4. ↑ Template:Cite book
  5. ↑ Template:Cite web
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