 TOPICS # 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).

Antipedal Triangle, Fagnano's Problem, Orthic Triangle, Pedal Circle, Pedal Line

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## References

Coxeter, H. S. M. and Greitzer, S. L. "Pedal Triangles." §1.9 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 22-26, 1967.Gallatly, W. "Pedal Triangles." Ch. 5 in The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, pp. 37-45, 1913.Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 67-74, 1995.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Stewart, B. M. "Cyclic Properties of Miquel Polygons." Amer. Math. Monthly 47, 462-466, 1940.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, 1991.

Pedal Triangle

## Cite this as:

Weisstein, Eric W. "Pedal Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PedalTriangle.html