Orthogonal Circles


Orthogonal circles are orthogonal curves, i.e., they cut one another at right angles. By the Pythagorean theorem, two circles of radii r_1 and r_2 whose centers are a distance d apart are orthogonal if


Two circles with Cartesian equations


are orthogonal if


A theorem of Euclid states that, for the orthogonal circles in the above diagram,


(Dixon 1991, p. 65).

The radical lines of three given circles concur in the radical center R. If a circle with center R cuts any one of the three circles orthogonally, it cuts all three orthogonally. This circle is called the orthogonal circle (or radical circle) of the system. The orthogonal circle is the locus of a point whose polars with respect to the three given circles are concurrent (Lachlan 1893, p. 237).

The following table lists circles orthogonal to various named circle.

See also

Circle, Midcircle, Monge's Problem, Radical Center, Radical Circle

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Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 42, 1888.Dixon, R. Mathographics. New York: Dover, pp. 65-66, 1991.Durell, C. V. "Orthogonal Circles." Ch. 8 in Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 88-92, 1928.Euclid. The Thirteen Books of the Elements, 2nd ed. unabridged, Vol. 3: Books X-XIII. New York: Dover, p. 36, 1956.Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, 1893.Pedoe, D. Circles: A Mathematical View, rev. ed. Washington, DC: Math. Assoc. Amer., p. xxiv, 1995.

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Orthogonal Circles

Cite this as:

Weisstein, Eric W. "Orthogonal Circles." From MathWorld--A Wolfram Web Resource.

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