Steiner point

In triangular geometry , the Steiner point is one of the excellent points of a flat triangle . The Steiner Point is a triangular center, which is designated X (99) in Clark Kimberling's Encyclopedia of Triangle Centers . Jakob Steiner (1796–1863), a Swiss mathematician, described the point in 1826. Steiner's name was given to the point in 1886 by Joseph Neuberg .

definition

The straight line through A parallel to B'C ' , the straight line through B parallel to C'A' and the straight line through C parallel to A'B ' intersect at the Steiner point.

The Steiner point can be defined as follows. (This is not Steiner's original definition.)

Let ABC be a given triangle. Furthermore, let O be the center of the circle and K the symmetry point of the triangle ABC . The circle with the diameter OK is the Brocard circle of triangle ABC . The straight line through O , which is perpendicular to the straight line BC , intersects the Brocard circle at a further point A ' . Correspondingly, the straight line through O , which is perpendicular to CA , intersects the Brocard circle at a further point B ' . The straight line through O , which is perpendicular to AB , intersects the Brocard circle at another point C ' . (The triangle A'B'C ' is also called the Brocard triangle of triangle ABC .) Now L _{A is} the parallel to B'C' through A , L _B the parallel to C'A ' through B and L _C die Parallel to A'B ' through C , the three straight lines L _A , L _B and L _C intersect at one point. The common point of intersection is the Steiner point of triangle ABC .

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

Alternative construction of the Steiner point

For the given triangle ABC, let O be the circumcenter and K be the symmetry point. Furthermore, let I _{A be} the mirror image (axis reflection) of the straight line OK with respect to the axis BC . Correspondingly, let l _{B be} the mirror image of straight line OK with respect to axis CA and l _{C be} the mirror image of straight line OK with respect to axis AB . Let the intersection of l _B and l _C be denoted by A ″ , the intersection of l _C and l _A with B ″ and the intersection of l _A and l _B with C ″ . It can be shown that the three straight lines AA ″ , BB ″ and CC ″ intersect at one point, namely at the Steiner point of the triangle ABC .

Trilinear and barycentric coordinates

Trilinear coordinates of the Steiner point are given by

${\displaystyle {\frac {bc}{b^{2}-c^{2}}}:{\frac {ca}{c^{2}-a^{2}}}:{\frac {ab}{a^{2}-b^{2}}}}$ or (equivalent)

${\displaystyle b^{2}c^{2}\csc(\beta -\gamma ):c^{2}a^{2}\csc(\gamma -\alpha ):a^{2}b^{2}\csc(\alpha -\beta )}$,

barycentric coordinates by

${\displaystyle {\frac {1}{b^{2}-c^{2}}}:{\frac {1}{c^{2}-a^{2}}}:{\frac {1}{a^{2}-b^{2}}}}$.

As usual, the terms , and stand for the side lengths and , and for the sizes of the corresponding angles. ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle \gamma }$

properties

1. If the given triangle is equilateral , the Steiner point is not defined. If the triangle is isosceles but not equilateral, the Steiner point coincides with the point, that is, with the corner where the two sides of equal length meet. In all other cases the Steiner point lies outside the triangle.
2. The Steiner point lies on the circumference of the triangle ABC .
3. The Steiner point lies on the Steiner umellipse of the triangle ABC , i.e. on the ellipse with the smallest area that goes through A , B and C.
4. The Simson line of the Steiner point of triangle ABC is parallel to the straight line OK , where O is the circumcenter and K is the point of symmetry of triangle ABC .

Tarry point

The perpendicular to B'C ' through A , the perpendicular to C'A' through B and the perpendicular to A'B ' through C intersect at the tarry point.

The Tarry point of a triangle - named after Gaston Tarry - is closely related to the Steiner point. For a given triangle ABC , the point on the circumference that is exactly opposite the Steiner point is called the Tarry point of the triangle. The Tarry Point is a triangular center that is labeled X (98) in the Encyclopedia of Triangle Centers. The trilinear coordinates of the Tarry point are given by the following (equivalent) expressions:

${\displaystyle \sec(\alpha +\omega ):\sec(\beta +\omega ):\sec(\gamma +\omega ),}$

wherein the Brocard angle of the triangle is ${\displaystyle \omega }$ ${\displaystyle ABC}$

${\displaystyle {\frac {bc}{b^{4}+c^{4}-a^{2}b^{2}-a^{2}c^{2}}}:{\frac {ca}{c^{4}+a^{4}-b^{2}c^{2}-b^{2}a^{2}}}:{\frac {ab}{a^{4}+b^{4}-c^{2}a^{2}-c^{2}b^{2}}}}$

${\displaystyle a}$, , While the lengths of the triangle sides, , , the sizes of the corresponding angle. ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle \gamma }$

The Tarry point can also be defined in the same way as the Steiner point:

Let ABC be a given triangle and A'B'C ' the corresponding Brocard triangle (see above). It should also be L _A solder to B'C ' by A , L _B , the solder to C'A' by B and L _C , the solder to A'B ' by C . Then the straight lines L _A , L _B and L _C intersect at a point. The common point is the Tarry point of triangle ABC .

Individual evidence

1. ^ Paul E. Black: Steiner point . In: Dictionary of Algorithms and Data Structures . US National Institute of Standards and Technology. Retrieved May 17, 2012.
2. ↑ ^{a b c} Clark Kimberling: Steiner point . Retrieved May 17, 2012.
3. ^ J. Neuberg: Sur le point de Steiner . In: Journal de mathématiques spéciales . 1886, p. 29.
4. ^ Tarry Point (MathWorld)