Given a point with trilinear coordinates 
, the anticevian triangle 
of a triangle 
with respect to 
is a triangle such that
1. 
passes through 
,
passes through 
,
and 
passes through 
.
2. 
,
,
and 
pass through 
.
3. 
is the Cevian triangle of 
with respect to 
.
The anticevian triangle has trilinear vertex matrix
![[-alpha beta gamma; alpha -beta gamma; alpha beta -gamma]](/images/equations/AnticevianTriangle/NumberedEquation1.svg) |
(1)
|
(Kimberling 1998, pp. 55 and 185), and is a central triangle of type 1 (Kimberling 1998, p. 55).
If 
is the Cevian triangle of 
and 
is an anticevian triangle, then 
and 
are harmonic conjugates
with respect to 
and 
.
The following table summarizes a number of special anticevian triangles for various special anticevian points 
, including their Kimberling center designations.
| anticevian point | Kimberling center | anticevian triangle |
incenter  |  | excentral triangle |
triangle
centroid  |  | anticomplementary triangle |
symmedian point  |  | tangential triangle |
The side lengths of the anticevian triangle with respect to an anticevian point 
are given by
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
The triangle area of the anticevian triangle 
for anticevian point with
trilinear coordinates 
is
 |
(5)
|
where 
is the triangle area of 
.
