Antipedal Triangle
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.gif) |
(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).
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