Pedal Circle


The pedal circle with respect to a pedal point P of a triangle DeltaA_1A_2A_3 is the circumcircle of the pedal triangle DeltaP_1P_2P_3 with respect to P. Amazingly, the vertices of the pedal triangle DeltaQ_1Q_2Q_3 of the isogonal conjugate point Q of P also lie on the same circle (Honsberger 1995). If the pedal point is taken as the incenter, the pedal circle is given by the incircle.

The radius of the pedal circle of a point P is


(Johnson 1929, p. 141).

When P is on a side of the triangle, the line between the two perpendiculars is called the pedal line. Given four points, no three of which are collinear, then the four pedal circles of each point for the triangle formed by the other three have a common point through which the nine-point circles of the four triangles pass.

See also

Fontené Theorems, Griffiths' Theorem, Miquel Point, Nine-Point Circle, Pedal Line, Pedal Triangle

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Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 50, 1971.Fontené, G. "Sur le cercle pédal." Nouv. Ann. Math. 65, 55-58, 1906.Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., p. 54, 1991.Honsberger, R. "The Pedal Circle." §7.4 (viii) in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 67-69, 1995.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.

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Pedal Circle

Cite this as:

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

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