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Radical Center


RadicalCenter

The radical lines of three circles are concurrent in a point known as the radical center (also called the power center). This theorem was originally demonstrated by Monge (Dörrie 1965, p. 153). It is a special case of the three conics theorem (Evelyn et al. 1974, pp. 13 and 15).

The point of concurrence of the three radical lines of three circles is the point

 |k k^' k^(''); m m^' m^(''); n n^' nn^'|:|k k^' k^(''); n n^' n^(''); l l^' l^('')|:|k k^' k^(''); l l^' l^(''); m m^' m^('')|

(Kimberling 1998, p. 225).


See also

Apollonius' Problem, Circle-Circle Intersection, Concurrent, Monge's Problem, Orthogonal Circles, Radical Circle, Radical Line, Three Conics Theorem

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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. Dublin: Hodges, Figgis, & Co., p. 43, 1888.Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 35, 1967.Dörrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, 1965.Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 125, 1928.Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. "The Three-Conics Theorem." §2.2 in The Seven Circles Theorem and Other New Theorems. London: Stacey International, pp. 11-18, 1974.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 32, 1929.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, p. 185, 1893.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 35, 1991.

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Radical Center

Cite this as:

Weisstein, Eric W. "Radical Center." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RadicalCenter.html

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