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
(1)
|
(Kimberling 1998, p. 186), and is a central triangle of type 2 (Kimberling 1998, p. 55).
The side lengths are
(2)
| |||
(3)
| |||
(4)
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where is the circumradius of , and area is
(5)
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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)
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(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)
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(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).