The circum-medial triangle 
is the circumcevian
triangle of a reference triangle 
with respect to the triangle
centroid 
of 
(Kimberling 1998, pp. 162-163).
It has trilinear vertex matrix
![[-abc c(b^2+c^2) b(b^2+c^2); c(c^2+a^2) -abc a(c^2+a^2); b(a^2+b^2) a(a^2+b^2) -abc].](/images/equations/Circum-MedialTriangle/NumberedEquation1.svg) |
Its area is
![Delta^'=((a^2+b^2+c^2)^3Delta)/([2(a^2+b^2)-c^2][2(a^2+c^2)-b^2][2(b^2+c^2)-a^2]),](/images/equations/Circum-MedialTriangle/NumberedEquation2.svg) |
where 
is the area of 
.
The following table gives the centers of the circum-medial triangle in terms of the centers of the reference triangle that correspond
to Kimberling centers 
.
 | center of circum-medial triangle |  | center of reference triangle |
 | circumcenter |  | circumcenter |
 | symmedian point |  | trilinear quotient  |
 |  |  | sr( ,  ) |
 | focus of Kiepert parabola |  | sr( ,  ) |
 | isogonal
conjugate of  |  | isogonal
conjugate of  |
 | isogonal
conjugate of  |  | odd  infinity point |
 | first Brocard-axis intercept of circumcircle |  | Steiner point |
 | second Brocard-axis intercept of circumcircle |  | Tarry point |
