Radical Line -- from Wolfram MathWorld
- ️Weisstein, Eric W.
The radical line, also called the radical axis, is the locus of points of equal circle power with respect to two nonconcentric circles. By the chordal theorem, it is perpendicular to the line of centers (Dörrie 1965).
Let the circles have radii 
and 
and their centers be separated by a distance 
. If the circles intersect
in two points, then the radical line is the line passing through the points of intersection.
If not, then draw any two circles which cut each original
circle twice. Draw lines through each pair of points of
intersection of each circle. The line connecting their
two points of intersection is then the radical line.
Given two circles with trilinear equations
 |
(1) |
their radical line has equation
 |
(2) |
(Kimberling 1998, p. 224).
The radical line is located at distances
along the line of centers from 
and 
, respectively, where
 |
(5) |
The radical line of any two polar circles is the altitude from the third vertex.
The following table gives the radical lines of pairs of circles that correspond to Kimberling centers
See also
Chordal Theorem, Circle-Circle Intersection, Circle Power, Coaxal Circles, Inverse Points, Inversion, Radical Center
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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. Introduction
to Geometry, 2nd ed. New York: Wiley, p. 86, 1969.Coxeter,
H. S. M. and Greitzer, S. L. "The Radical Axis of Two Circles."
§2.2 in Geometry
Revisited. Washington, DC: Math. Assoc. Amer., pp. 31-34, 1967.Dixon,
R. Mathographics.
New York: Dover, p. 68, 1991.Dörrie, H. 100
Great Problems of Elementary Mathematics: Their History and Solutions. New
York: Dover, p. 153, 1965.Gallatly, W. "The Radical Axis of
and 
." §23 in The
Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 16, 1913.Durell,
C. V. Modern
Geometry: The Straight Line and Circle. London: Macmillan, p. 121, 1928.Johnson,
R. A. Modern
Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle.
Boston, MA: Houghton Mifflin, pp. 28-34 and 176-177, 1929.Kimberling,
C. "Triangle Centers and Central Triangles." Congr. Numer. 129,
1-295, 1998.Lachlan, R. "The Radical Axis of Two Circles."
§304-312 in An
Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 185-189,
1893.Wells, D. The
Penguin Dictionary of Curious and Interesting Geometry. London: Penguin,
p. 35, 1991.
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Cite this as:
Weisstein, Eric W. "Radical Line." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RadicalLine.html