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
(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.
antipedal point | Kimberling center | antipedal triangle |
incenter | excentral triangle | |
circumcenter | tangential triangle | |
orthocenter | anticomplementary triangle |
The antipedal triangle with respect to and has side lengths
(2)
| |||
(3)
| |||
(4)
|
where is the circumradius of , and area
(5)
|
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).