Steiner point (triangle)

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

{{legend-line|solid #B8860B 2px|Triangle {{mvar|ABC}}}}

{{legend-line|solid #8B0000 2px|Triangle {{mvar|A'B'C'}} (Brocard triangle of {{mvar|ABC}})}}

{{legend-line|solid #00BFFF 2px|Circumcircle of triangle {{mvar|ABC}}, centered at {{mvar|O}}}}

{{legend-line|solid magenta 2px|Brocard circle of triangle {{mvar|ABC}}}}

Lines concurring at the Steiner point:

{{legend-line|solid green 2px|{{mvar|L{{sub|A}}}}: line through {{mvar|A}} parallel to {{mvar|B'C'}}}}

{{legend-line|solid blue 2px|{{mvar|L{{sub|B}}}}: line through {{mvar|B}} parallel to {{mvar|C'A'}}}}

{{legend-line|solid red 2px|{{mvar|L{{sub|C}}}}: line through {{mvar|C}} parallel to {{mvar|A'B'}}}}

]]

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

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

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

File:Steiner point Construction 02.svg

:Let {{mvar|ABC}} be any given triangle. Let {{mvar|O}} be the circumcenter and {{mvar|K}} be the symmedian point of triangle {{mvar|ABC}}. Let {{mvar|l{{sub|A}}}} be the reflection of the line {{mvar|OK}} in the line {{mvar|BC}}, {{mvar|l{{sub|B}}}} be the reflection of the line {{mvar|OK}} in the line {{mvar|CA}} and {{mvar|l{{sub|C}}}} be the reflection of the line {{mvar|OK}} in the line {{mvar|AB}}. Let the lines {{mvar|l{{sub|B}}}} and {{mvar|l{{sub|C}}}} intersect at {{mvar|A″}}, the lines {{mvar|l{{sub|C}}}} and {{mvar|l{{sub|A}}}} intersect at {{mvar|B″}} and the lines {{mvar|l{{sub|A}}}} and {{mvar|l{{sub|B}}}} intersect at {{mvar|C″}}. Then the lines {{mvar|AA″}}, {{mvar|BB″}} and {{mvar|CC″}} are concurrent. The point of concurrency is the Steiner point of triangle {{mvar|ABC}}.

The trilinear coordinates of the Steiner point are given below.

:{{tmath|1= bc / (b^2 - c^2) : ca / (c^2 - a^2) : ab / (a^2 - b^2) }}

:{{tmath|{{=}} b^2 c^2 \csc(b - C) : c^2 a^2 \csc(c - a) : a^2 b^2 \csc(a - b) }}

1. The Steiner circumellipse of triangle {{mvar|ABC}}, also called the Steiner ellipse, is the ellipse of least area that passes through the vertices {{mvar|A}}, {{mvar|B}} and {{mvar|C}}. The Steiner point of triangle {{mvar|ABC}} lies on the Steiner circumellipse of triangle {{mvar|ABC}}.
2. The Simson line of the Steiner point of triangle {{mvar|ABC}} is parallel to the line {{mvar|OK}} where {{mvar|O}} is the circumcenter and {{mvar|K}} is the symmmedian point of triangle {{mvar|ABC}}.
3. The Steiner point of triangle {{mvar|ABC}} is the Brianchon point of the Kiepert parabola with respect to triangle {{mvar|ABC}}.{{cite journal |last1=Eddy |first1=R. H. |last2=Fritsch |first2=R. |title=The Conics of Ludwig Kiepert: A Comprehensive Lesson in the Geometry of the Triangle |journal=Math. Mag. |date=1994 |volume=67 |issue=3 |pages=188–205|doi=10.1080/0025570X.1994.11996212 }}

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.{{cite book|last=Honsberger|first=Ross|title=Episodes in nineteenth and twentieth century Euclidean geometry|year=1965|publisher=The Mathematical Association of America|pages=119–124}} 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 $\backslash left(\backslash frac\{\backslash pi\; -\; A\}\{a\}\; :\; \backslash frac\{\backslash pi\; -\; B\}\{b\}\; :\; \backslash frac\{\backslash pi\; -\; C\}\{c\}\backslash right)$.{{cite web|last=Eric W.|first=Weisstein|title=Steiner Curvature Centroid|url=http://mathworld.wolfram.com/SteinerCurvatureCentroid.html|publisher=MathWorld—A Wolfram Web Resource.|accessdate=17 May 2012}} It is the triangle center designated as X(1115) in Encyclopedia of Triangle Centers.

File:Tarry point Construction.svg

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

:{{tmath|1= \sec(A + \omega) : \sec(B + \omega) : \sec(C + \omega) = f(a,b,c) : f(b,c,a) : f(c,a,b)}}

:::where {{mvar|ω}} is the Brocard angle of triangle {{mvar|ABC}}

:::and {{tmath|1=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 {{mvar|ABC}} be any given triangle. Let {{mvar|A'B'C'}} be the Brocard triangle of triangle {{mvar|ABC}}. Let {{mvar|L{{sub|A}}}} be the line through {{mvar|A}} perpendicular to the line {{mvar|B'C'}}, {{mvar|L{{sub|B}}}} be the line through {{mvar|B}} perpendicular to the line {{mvar|C'A'}} and {{mvar|L{{sub|C}}}} be the line through {{mvar|C}} perpendicular to the line {{mvar|A'B'}}. Then the three lines {{mvar|L{{sub|A}}}}, {{mvar|L{{sub|B}}}} and {{mvar|L{{sub|C}}}} are concurrent. The point of concurrency is the Tarry point of triangle {{mvar|ABC}}.

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Category:Triangle centers