The circumcevian triangle 
of a point 
with respect to a triangle 
has vertices
at the second intersections of the lines 
, 
, and 
with the circumcircle of
,
respectively (Kimberling 1998, p. 201). Configurations in which an intersection
is undefined or the resulting triangle is degenerate
are excluded.
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). When both triangles are defined and
nondegenerate, their corresponding sides are parallel iff 
lies on the M'Cay
cubic. When the resulting homothecy has a finite
homothetic center, that center lies on the Lemoine cubic (Gibert).
