Pedal Triangle

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)

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).

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