The radical center, also called the power center, of three circles with noncollinear centers is the unique point at which
their pairwise radical lines are concurrent .
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).
For a nondegenerate reference triangle whose sides have lengths circle in trilinear
coordinates as
with primed and double-primed coefficients for the other two circles . The radical center then has trilinear
coordinates
(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. Coxeter, H. S. M.
and Greitzer, S. L. Geometry
Revisited. Dörrie,
H. 100
Great Problems of Elementary Mathematics: Their History and Solutions. Durell, C. V. Modern
Geometry: The Straight Line and Circle. 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. Johnson, R. A. Modern
Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Kimberling, C. "Triangle
Centers and Central Triangles." Congr. Numer. 129 , 1-295, 1998. Lachlan,
R. An
Elementary Treatise on Modern Pure Geometry. Wells, D. The
Penguin Dictionary of Curious and Interesting Geometry. Referenced on Wolfram|Alpha Radical Center
Cite this as:
Weisstein, Eric W. "Radical Center." From
MathWorld https://mathworld.wolfram.com/RadicalCenter.html
Subject classifications
, 
,
and 
,
write the first circle in trilinear
coordinates as
