Coaxal Circles
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 
. Picking 
then
gives the two circles
of zero radius, known as point circles. The two point circles 
,
real or imaginary, are called the limiting points.
SEE ALSO: Circle,
Coaxaloid System,
Gauss-Bodenmiller Theorem,
Limiting Point,
Point
Circle,
Radical Line
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. Dublin: Hodges,
Figgis, & Co., pp. 113-126, 1888.
Coolidge, J. L. "Coaxal Circles." §1.7 in A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, pp. 95-113,
1971.
Coxeter, H. S. M. and Greitzer, S. L. "Coaxal Circles." §2.3 in Geometry
Revisited. Washington, DC: Math. Assoc. Amer., pp. 35-36 and 122, 1967.
Dixon, R. Mathographics.
New York: Dover, pp. 68-72, 1991.
Durell, C. V. "Coaxal Circles." Ch. 11 in Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 121-125,
1928.
Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle.
Boston, MA: Houghton Mifflin, pp. 34-37, 199, and 279, 1929.
Lachlan, R. "Coaxal Circles." Ch. 13 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 199-217,
1893.
Steinhaus, H. Mathematical
Snapshots, 3rd ed. New York: Dover, pp. 143-144, 1999.
Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin,
pp. 33-34, 1991.
Referenced on Wolfram|Alpha:
Coaxal Circles
CITE THIS AS:
Weisstein, Eric W. "Coaxal Circles." From
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