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Radical Line


RadicalAxis

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 r_1 and r_2 and their centers be separated by a distance d. 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

 (lalpha+mbeta+ngamma)(aalpha+bbeta+cgamma) 
 +k(abetagamma+bgammaalpha+calphagamma)=0  
(l^'alpha+m^'beta+n^'gamma)(aalpha+bbeta+cgamma) 
 +k^'(abetagamma+bgammaalpha+calphabeta)=0,
(1)

their radical line has equation

 (k^'l-kl^')alpha+(k^'m-km^')beta+(k^'n-kn^')gamma=0
(2)

(Kimberling 1998, p. 224).

The radical line is located at distances

d_1=(d^2+r_1^2-r_2^2)/(2d)
(3)
d_2=-(d^2+r_2^2-r_1^2)/(2d)
(4)

along the line of centers from C_1 and C_2, respectively, where

 d=d_1-d_2.
(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

circle 1circle 2lineline name
anticomplementary circleBevan circleL_(1914)
anticomplementary circleBrocard circleL_(251)
anticomplementary circlecircumcircleL_(32)de Longchamps line
anticomplementary circleConway circleL_(41)
anticomplementary circlede Longchamps circleL_(32)de Longchamps line
anticomplementary circleFuhrmann circleL_(2300)
anticomplementary circleStammler circleL_(50)
Apollonius circlecircumcircleL_(940)
Apollonius circleexcircles radical circleL_1antiorthic axis
Apollonius circlenine-point circleL_1antiorthic axis
Bevan circlecircumcircleL_1antiorthic axis
Bevan circleConway circleL_(2176)
Bevan circlede Longchamps circleL_(172)
Bevan circleDou circleL_(1069)
Bevan circleexcircles radical circleL_(55)Gergonne line
Bevan circleextangents circleL_(581)
Bevan circleFuhrmann circleL_(284)
Bevan circlepolar circleL_(55)Gergonne line
Brocard circlecircumcircleL_2Lemoine axis
Brocard circlecosine circleL_(193)
Brocard circlede Longchamps circleL_(1627)
Brocard circleLucas circles radical circleL_2Lemoine axis
Brocard circleLucas inner circleL_2Lemoine axis
Brocard circlenine-point circleL_(25)
Brocard circleorthocentroidal circleL_(23)
Brocard circleorthoptic circle of the Steiner inellipseL_(1995)
Brocard circlepolar circleL_(22)
Brocard circlesecond Brocard circleL_(385)
circumcircleConway circleL_(213)
circumcirclecosine circleL_(69)
circumcirclede Longchamps circleL_(32)de Longchamps line
circumcircleDou circleL_(155)
circumcircleexcircles radical circleL_(56)
circumcirclefirst Lemoine circleL_(141)
circumcircleFuhrmann circleL_(48)
circumcircleGallatly circleL_(183)
circumcircleincentral circleL_(191)
circumcircleincircleL_(220)
circumcircleinner Napoleon triangleL_(15)
circumcircleLucas circles radical circleL_2Lemoine axis
circumcircleLucas inner circleL_2Lemoine axis
circumcircleMandart circleL_(221)
circumcircleMoses circleL_(599)
circumcirclenine-point circleL_3orthic axis
circumcircleorthocentroidal circleL_3orthic axis
circumcircleorthoptic circle of the Steiner inellipseL_3orthic axis
circumcircleouter Napoleon circleL_(16)
circumcircleParry circleL_(690)
circumcirclepolar circleL_3orthic axis
circumcirclereflection circleL_(195)
circumcircleStammler circles radical circleL_(3003)
circumcircleStevanovic circleL_(905)
circumcirclesymmedial circleL_(2896)
circumcircletangential circleL_3orthic axis
circumcircleTaylor circleL_(394)
Conway circlede Longchamps circleL_(31)
cosine circlefirst Lemoine circleL_(524)
de Longchamps circleincircleL_(3052)
de Longchamps circleNeuberg circles radical circleL_(1613)
de Longchamps circlenine-point circleL_(1384)
de Longchamps circlepolar circleL_(3053)
de Longchamps circlesecond Steiner circleL_(512)
de Longchamps circleYff contact circleL_(649)Nagel line
Dou circlepolar circleL_(1147)
Euler-Gergonne-Soddy circleGEOS circleL_(657)Soddy line
excircles radical circleMandart circleL_(603)
excircles radical circlenine-point circleL_1antiorthic axis
excircles radical circlepolar circleL_(55)Gergonne line
first Droz-Farny circlenine-point circleL_(1609)
first Droz-Farny circlesecond Droz-Farny circleL_(50)
first Lemoine circlenine-point circleL_(159)
Fuhrmann circlepolar circleL_(219)
half-altitude circlenine-point circleL_(64)
half-altitude circlepolar circleL_(154)
incircleinner Soddy circleL_(55)Gergonne line
incirclemixtilinear incircles radical circleL_(2256)
incirclenine-point circleL_(101)
incircleouter Soddy circleL_(55)Gergonne line
inner Napoleon circleouter Napoleon circleL_(187)
inner Soddy circleouter Soddy circleL_(55)Gergonne line
inner Vecten circleouter Vecten circleL_(51)
Lester circleorthocentroidal circleL_(1510)
Lucas circles radical circleLucas inner circleL_2Lemoine axis
Mandart circlenine-point circleL_(109)
mixtilinear incircles radical circlepolar circleL_(1604)
Moses circlenine-point circleL_(1634)
nine-point circleorthocentroidal circleL_3orthic axis
nine-point circleorthoptic circle of the Steiner inellipseL_3orthic axis
nine-point circlepolar circleL_3orthic axis
nine-point circleSpieker circleL_(1616)
nine-point circletangential circleL_3orthic axis
nine-point circleTaylor circleL_(155)
nine-point circlefirst Droz-Farny circleL_(1609)
orthocentroidal circleorthoptic circle of the Steiner inellipseL_3orthic axis
orthocentroidal circlepolar circleL_3orthic axis
orthocentroidal circlereflection circleL_(1493)
orthocentroidal circletangential circleL_3orthic axis
orthoptic circle of the Steiner inellipsepolar circleL_3orthic axis
orthoptic circle of the Steiner inellipsetangential circleL_3orthic axis
polar circlesecond Droz-Farny circleL_(577)
polar circletangential circleL_3orthic axis
polar circleTaylor circleL_(1181)
Stammler circleStammler circles radical circleL_(39)
Stammler circles radical circlefirst Droz-Farny circleL_(571)
Stammler circles radical circlesecond Droz-Farny circleL_(571)

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 O(R) and I(r)." §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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Radical Line

Cite this as:

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

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