Given a triangle 
and a point 
not a vertex of 
, define the 
-vertex of the circumcevian triangle as the point other than
in which the line 
meets the circumcircle of 
, and similarly for 
and 
. Then 
is called the circumcevian triangle of 
(Kimberling 1998, p. 201).
The circumcevian triangle 
with respect to the point 
has trilinear
vertex matrix
![[-abetagamma (bgamma+cbeta)beta (bgamma+cbeta)gamma; (calpha+agamma)alpha -bgammaalpha (calpha+agamma)gamma; (abeta+balpha)alpha (abeta+balpha)beta -calphabeta]](/images/equations/CircumcevianTriangle/NumberedEquation1.svg) |
(1)
|
and area
 |
(2)
|
where 
is the area of 
and
 |
(3)
|
Circumcevian triangles for various choices of 
are summarized in the table below.
| Kimberling | circumcevian point  | circumcevian triangle |
 | incenter  | circumcircle mid-arc triangle |
 | triangle
centroid  | circum-medial triangle |
 | orthocenter  | circum-orthic triangle |
Every triangle inscribed in the circumcircle of a reference triangle 
is congruent to exactly one circumcevian triangle of
(Kimberling 2005).
The circumcevian triangle of 
is similar to the pedal triangle
of 
(Kimberling 1998), and it is homothetic to 
iff 
lies on the M'Cay cubic. The
homothetic center lies on the Lemoine
cubic (Gibert).
