Pedal Triangle


Given a point P, the pedal triangle of P is the triangle whose polygon vertices are the feet of the perpendiculars from P to the side lines. The pedal triangle of a triangle with trilinear coordinates alpha:beta:gamma and angles A, B, and C has trilinear vertex matrix

 [0 beta+alphacosC gamma+alphacosB; alpha+betacosC 0 gamma+betacosA; alpha+gammacosB beta+gammacosA 0]

(Kimberling 1998, p. 186), and is a central triangle of type 2 (Kimberling 1998, p. 55).

The side lengths are


where R is the circumradius of DeltaABC, and area is


where Delta is the area of DeltaABC.

The following table summarizes a number of special pedal triangles for various special pedal points P.

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 nth pedal n-gon of any n-gon is similar to the original one. It is also true that


(Johnson 1929, pp. 135-136; Stewart 1940; Coxeter and Greitzer 1967, p. 25). The area Delta_P of the pedal triangle of a point P is proportional to the power of P with respect to the circumcircle,


(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).

See also

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

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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.

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Pedal Triangle

Cite this as:

Weisstein, Eric W. "Pedal Triangle." From MathWorld--A Wolfram Web Resource.

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