triangle geometry A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collin ...

, the Steiner point is a particular point associated with a

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

. It is a

triangle center In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. For exampl ...

and it is designated as the center X(99) in

Clark Kimberling Clark Kimberling (born November 7, 1942 in Hinsdale, Illinois) is a mathematician, musician, and composer. He has been a mathematics professor since 1970 at the University of Evansville. His research interests include triangle centers, integer seq ...

Encyclopedia of Triangle Centers The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or " centers" associated with the geometry of a triangle. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville. , the ...

Jakob Steiner Jakob Steiner (18 March 1796 – 1 April 1863) was a Swiss mathematician who worked primarily in geometry. Life Steiner was born in the village of Utzenstorf, Canton of Bern. At 18, he became a pupil of Heinrich Pestalozzi and afterwards st ...

(1796–1863), 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. (This is not the way in which Steiner defined it.) :Let be any given triangle. Let be the

circumcenter In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

and be the

symmedian point In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the co ...

of triangle . The

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...

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 In geometry, the Brocard triangle of a triangle is a triangle formed by the intersection of lines from a vertex to its corresponding Brocard point and a line from another vertex to its corresponding Brocard point and the other two points construc ...

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 Concurrent means happening at the same time. Concurrency, concurrent, or concurrence may refer to: Law * Concurrence, in jurisprudence, the need to prove both ''actus reus'' and ''mens rea'' * Concurring opinion (also called a "concurrence"), a ...

. The point of concurrency is the ''Steiner point'' of triangle . In the

the Steiner point is defined as follows; Steiner point Construction 02

:Let be any given triangle. Let be the

and be the

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 In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is ...

of the Steiner point are given below. : :

Properties

#The

Steiner circumellipse In geometry, the Steiner ellipse of a triangle, also called the Steiner circumellipse to distinguish it from the Steiner inellipse, is the unique circumellipse (ellipse that touches the triangle at its vertex (geometry), vertices) whose center is ...

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.'' 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 : \frac : \frac\right)

. It is the triangle center designated as X(1115) in

. #The

Simson line In geometry, given a triangle and a point on its circumcircle, the three closest points to on lines , , and are collinear. The line through these points is the Simson line of , named for Robert Simson. The concept was first published, howeve ...

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 In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

of triangle diametrically opposite to the Steiner point of triangle is called the

Tarry point In geometry, the Tarry point for a triangle is a point of concurrency of the lines through the vertices of the triangle perpendicular to the corresponding sides of the triangle's first Brocard triangle . The Tarry point lies on the other ...

of triangle . The Tarry point is a triangle center and it is designated as the center X(98) in

. 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

. The point of concurrency is the ''Tarry point'' of triangle .

References

{{reflist Triangle centers