The pedal triangle of a point 
with respect to a reference
triangle 
is the triangle whose polygon
vertices are the feet of the perpendiculars
from 
to the side lines. For a point with trilinear
coordinates 
,
its trilinear vertex matrix is
![[0 beta+alphacosC gamma+alphacosB; alpha+betacosC 0 gamma+betacosA; alpha+gammacosB beta+gammacosA 0]](/images/equations/PedalTriangle/NumberedEquation1.svg) |
(1)
|
(Kimberling 1998, p. 186), where 
, 
,
and 
are the angles of the reference triangle. The
pedal triangle is a central triangle of type
2 (Kimberling 1998, p. 55).
The side lengths are
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
where 
is the circumradius of 
, and area is
 |
(5)
|
where 
is the area of 
.
The following table summarizes a number of special pedal triangles for various special pedal points 
.
| pedal point | Kimberling center | pedal triangle |
incenter  |  | contact triangle |
circumcenter  |  | medial triangle |
orthocenter  |  | orthic triangle |
Bevan
point  |  | extouch triangle |
The symmedian point of a triangle is the triangle centroid of its pedal triangle (Honsberger 1995, pp. 72-74).
The third pedal triangle is similar to the original one. This theorem can be generalized to: the 
th pedal 
-gon of any 
-gon is similar to the original
one. It is also true that
 |
(6)
|
(Johnson 1929, pp. 135-136; Stewart 1940; Coxeter and Greitzer 1967, p. 25). The area 
of the pedal triangle of a point 
is proportional to the power
of 
with respect to the circumcircle,
 |  |  |
(7)
|
 |  |  |
(8)
|
(Johnson 1929, pp. 139-141).
The only closed billiards path of a single circuit in an acute triangle is the orthic triangle, the pedal triangle of the orthocenter. There are infinitely many multiple-circuit paths with segments parallel to the sides of the orthic triangle (Wells 1991).
