Any one of the eight Apollonius circles of three given circles is tangent to
a circle 
known as a Hart circle, as
are the other three Apollonius circles having
(1) like contact with two of the given circles and (2)
unlike contact with the third.
Hart's Theorem
See also
Apollonius Circle, Hart CircleExplore with Wolfram|Alpha
References
Casey, J. "On the Equations and Properties--(1) of the System of Circles Touching Three Circles in a Plane; (2) of the System of Spheres Touching Four Spheres in Space; (3) of the System of Circles Touching Three Circles on a Sphere; (4) of the System of Conics Inscribed to a Conic, and Touching Three Inscribed Conics in a Plane." Proc. Roy. Irish Acad. 9, 396-423, 1864-1866.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., pp. 106-107, 1888.Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 43, 1971.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 127-128, 1929.Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 254-257, 1893.Larmor, A. "Contacts of Systems of Circles." Proc. London Math. Soc. 23, 136-157, 1891.Referenced on Wolfram|Alpha
Hart's TheoremCite this as:
Weisstein, Eric W. "Hart's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HartsTheorem.html