The antipedal triangle 
of a reference
triangle 
with respect to a given point 
is the triangle of which 
is the pedal
triangle with respect to 
. If the point 
has trilinear coordinates 
and the angles
of 
are 
,
,
and 
,
then the antipedal triangle has trilinear vertex
matrix
![[-(beta+alphacosC)(gamma+alphacosB) (gamma+alphacosB)(alpha+betacosC) (beta+alphacosC)(alpha+gammacosB); (gamma+betacosA)(beta+alphacosC) -(gamma+betacosA)(alpha+betacosC) (alpha+betacosC)(beta+gammacosA); (beta+gammacosA)(gamma+alphacosB) (alpha+gammacosB)(gamma+betacosA) -(alpha+gammacosB)(beta+gammacosA)]](/images/equations/AntipedalTriangle/NumberedEquation1.svg) |
(1)
|
(Kimberling 1998, p. 187).
The antipedal triangle is a central triangle of type 2 (Kimberling 1998, p. 55).
The following table summarizes some named antipedal triangles with respect to special antipedal points. The antipedal triangle of the first Fermat point is an (apparently unnamed) equilateral triangle (Shenghui Yang, pers. comm. to E. Pegg, Jr., Jan. 3, 2025), which has side lengths
![a^'=b^'=c^'=2^(3/2)3^(-1/4)(12Delta(a^2+b^2+c^2)+4sqrt(3)(a^2b^2+b^2c^2+c^2a^2)-sqrt(3)(a^4+b^4+c^4))/([12Delta+sqrt(3)(a^2+b^2+c^2)]^(3/2)),](/images/equations/AntipedalTriangle/NumberedEquation2.svg) |
(2)
|
where 
are the side lengths of the reference triangle and 
its area (E. Weisstein, Jan. 6, 2025).
| antipedal point | Kimberling center | antipedal triangle |
incenter  |  | excentral triangle |
circumcenter  |  | tangential triangle |
orthocenter  |  | anticomplementary triangle |
first
Fermat point  |  | (unnamed?) equilateral triangle |
The antipedal triangle with respect to 
and 
has side lengths
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
where 
is the circumradius of 
, and area
 |
(6)
|
The isogonal conjugate of the antipedal triangle of a given triangle 
with respect to a point 
is the antipedal triangle of 
with respect to the isogonal
conjugate of 
.
It is also homothetic with the pedal triangle of
with respect to 
.
Furthermore, the product of the areas of the two homothetic triangles equals the
square of the area of the original triangle (Gallatly 1913, pp. 56-58).
