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Coaxal Circles


CoaxalCircles

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.

PointCircles

Members of a coaxal system satisfy

 x^2+y^2+2lambdax+c=(x+lambda)^2+y^2+c-lambda^2=0

for values of lambda. Picking lambda^2=c then gives the two circles

 (x+/-sqrt(c))^2+y^2=0

of zero radius, known as point circles. The two point circles (+/-sqrt(c),0), real or imaginary, are called the 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. 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.

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Coaxal Circles

Cite this as:

Weisstein, Eric W. "Coaxal Circles." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CoaxalCircles.html

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