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
(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 .