Given a point 
,
the pedal triangle of 
is the triangle whose polygon
vertices are the feet of the perpendiculars from 
to the side lines. The pedal triangle of a triangle
with trilinear coordinates 
and angles 
, 
,
and 
has trilinear vertex matrix
![[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), and 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 | anticevian 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 pedal triangle. There are an infinite number of multiple-circuit paths, but all segments are parallel to the sides of the pedal triangle (Wells 1991).
