Coaxal circles are circles whose centers are collinear and that share a common radical line . The collection
of all coaxal circles is called a pencil of coaxal circles (Coxeter and Greitzer
1967, p. 35). It is possible to combine the two types of coaxal systems illustrated
above such that the sets are orthogonal.
Note that not all circles sharing the same radical line need be coaxal, since the lines of their centers need only be perpendicular to the
radical line and therefore may not coincide.
Members of a coaxal system satisfy
for values of
of zero radius , known as point circles . The two point circles limiting
points .
See also Circle ,
Coaxaloid System ,
Gauss-Bodenmiller Theorem ,
Limiting Point ,
Point
Circle ,
Radical Line
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References Casey, J. "Coaxal Circles." §6.5 in 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. Coolidge, J. L. "Coaxal
Circles." §1.7 in A
Treatise on the Geometry of the Circle and Sphere. Coxeter, H. S. M. and Greitzer, S. L. "Coaxal
Circles." §2.3 in Geometry
Revisited. Dixon,
R. Mathographics. Durell, C. V. "Coaxal
Circles." Ch. 11 in Modern
Geometry: The Straight Line and Circle. Johnson, R. A. Modern
Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Lachlan,
R. "Coaxal Circles." Ch. 13 in An
Elementary Treatise on Modern Pure Geometry. Steinhaus, H. Mathematical
Snapshots, 3rd ed. Wells,
D. The
Penguin Dictionary of Curious and Interesting Geometry. Referenced on Wolfram|Alpha Coaxal Circles
Cite this as:
Weisstein, Eric W. "Coaxal Circles." From
MathWorld https://mathworld.wolfram.com/CoaxalCircles.html
Subject classifications
.
Picking 
then gives the two circles
, real or imaginary, are called the limiting
points.
