The Steiner triangle 
(a term coined here for the first time), is the Cevian
triangle of the Steiner point 
.
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].](/images/equations/SteinerTriangle/NumberedEquation1.svg) |
The vertices are the points of contact of the Kiepert parabola with the sidelines of the reference triangle.
It has area
 |
where 
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 
,
where 
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 
of the reference triangle, which is 
of the Steiner triangle (P. Moses, pers. comm., Dec. 31,
2004).
The circumcircle of the Steiner triangle is the second Steiner circle.
