TOPICS
Search

Steiner Triangle


SteinerTriangle

The Steiner triangle DeltaS_AS_BS_C (a term coined here for the first time), is the Cevian triangle of the Steiner point S.

It is the polar triangle of the Kiepert parabola.

It has the trilinear vertex matrix

 [0 (b^2-a^2)c b(a^2-c^2); (b^2-a^2)c 0 a(c^2-b^2); b(c^2-a^2) a(b^2-c^2) 0].

The vertices are the points of contact of the Kiepert parabola with the sidelines of the reference triangle.

It has area

 Delta^'=-2Delta,

where Delta is the area of the reference triangle and the negative sign indicates that the orientation is reversed. This is a special case of the general result that area of the Cevian triangle of any point on the Steiner circumellipse is -2Delta, where Delta is the area of the reference triangle.

The orthocenter of the Steiner triangle is the circumcenter of the reference triangle.

The Euler lines of the Steiner triangle and the reference triangle intersect at X_3 of the reference triangle, which is X_4 of the Steiner triangle (P. Moses, pers. comm., Dec. 31, 2004).

The circumcircle of the Steiner triangle is the second Steiner circle.


See also

Kiepert Parabola, Second Steiner Circle, Steiner Points

Explore with Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Steiner Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SteinerTriangle.html

Subject classifications