Orthogonal circles are orthogonal curves , i.e., they cut one another at right angles . By the Pythagorean
theorem , two circles of radii
(1)
Two circles with Cartesian equations
are orthogonal if
(4)
A theorem of Euclid states that, for the orthogonal circles in the above diagram,
(5)
(Dixon 1991, p. 65).
The radical lines of three given circles concur in the radical center 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.
circle orthogonal circle(s) Apollonius circle Stevanović circle Bevan circle Stevanović circle Brocard circle Parry circle circumcircle Parry
circle , Stevanović circle excircles radical
circle Stevanović
circle Lester
circle orthocentroidal
circle Lucas
circles radical circle Parry
circle nine-point
circle Stevanović
circle orthocentroidal
circle Lester circle , Stevanović circle orthoptic circle
of the Steiner inellipse polar
circle , Stevanović circle Parry circle Brocard circle , circumcircle ,
Lucas circles radical circle , Lucas
inner circle polar
circle second Droz-Farny
circle , Stevanović circle second Droz-Farny circle polar circle Stevanović circle Apollonius circle , Bevan
circle , circumcircle , excircles
radical circle , nine-point circle , orthocentroidal
circle , orthoptic circle
of the Steiner inellipse , polar circle , tangential
circle tangential
circle Stevanović
circle
See also Circle ,
Midcircle ,
Monge's Problem ,
Radical
Center ,
Radical Circle
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References 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. Dixon, R. Mathographics. Durell, C. V. "Orthogonal
Circles." Ch. 8 in Modern
Geometry: The Straight Line and Circle. Euclid. The
Thirteen Books of the Elements, 2nd ed. unabridged, Vol. 3: Books X-XIII. Lachlan, R. An
Elementary Treatise on Modern Pure Geometry. Pedoe,
D. Circles:
A Mathematical View, rev. ed. Referenced on Wolfram|Alpha Orthogonal Circles
Cite this as:
Weisstein, Eric W. "Orthogonal Circles."
From MathWorld https://mathworld.wolfram.com/OrthogonalCircles.html
Subject classifications
and 
whose centers are a distance 
apart are orthogonal if
. If a circle with center 
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).
