Difference between revisions of "Radical axis"
From Encyclopedia of Mathematics
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The set of points in the plane that, relative to two non-concentric circles | The set of points in the plane that, relative to two non-concentric circles | ||
| − | + | $$x^2+y^2-2a_1x-2b_1y-2c_1=0,$$ | |
| − | + | $$x^2+y^2-2a_2x-2b_2y-2c_2=0,$$ | |
| − | are points of the same power (cf. [[ | + | are points of the same power (cf. [[Degree of a point]]). The equation of the radical axis is: |
| − | + | $$(a_2-a_1)x+(b_2-b_1)y+(c_2-c_1)=0.$$ | |
The radical axis of two disjoint circles lies outside the circles and is perpendicular to the line through their centres (it is sometimes assumed that the radical axis of concentric circles is the line at infinity). The axis of two intersecting circles is the line passing through the points of intersection, and the radical axis of two touching circles is their common tangent. For any three circles with non-collinear centres the radical axes of each pair of circles pass through one point (the radical centre). | The radical axis of two disjoint circles lies outside the circles and is perpendicular to the line through their centres (it is sometimes assumed that the radical axis of concentric circles is the line at infinity). The axis of two intersecting circles is the line passing through the points of intersection, and the radical axis of two touching circles is their common tangent. For any three circles with non-collinear centres the radical axes of each pair of circles pass through one point (the radical centre). | ||
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Figure: r077090a | Figure: r077090a | ||
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====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L. Gaultier, ''J. de l'École Polytechn.'' , '''16''' (1813) pp. 147</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Berger, "Geometry" , '''I''' , Springer (1987)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> L. Gaultier, ''J. de l'École Polytechn.'' , '''16''' (1813) pp. 147</TD></TR> | ||
| + | <TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Berger, "Geometry" , '''I''' , Springer (1987)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963)</TD></TR> | ||
| + | </table> | ||
Latest revision as of 06:14, 16 April 2023
The set of points in the plane that, relative to two non-concentric circles
$$x^2+y^2-2a_1x-2b_1y-2c_1=0,$$
$$x^2+y^2-2a_2x-2b_2y-2c_2=0,$$
are points of the same power (cf. Degree of a point). The equation of the radical axis is:
$$(a_2-a_1)x+(b_2-b_1)y+(c_2-c_1)=0.$$
The radical axis of two disjoint circles lies outside the circles and is perpendicular to the line through their centres (it is sometimes assumed that the radical axis of concentric circles is the line at infinity). The axis of two intersecting circles is the line passing through the points of intersection, and the radical axis of two touching circles is their common tangent. For any three circles with non-collinear centres the radical axes of each pair of circles pass through one point (the radical centre).
Figure: r077090a
References
| [a1] | L. Gaultier, J. de l'École Polytechn. , 16 (1813) pp. 147 |
| [a2] | M. Berger, "Geometry" , I , Springer (1987) |
| [a3] | H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963) |
How to Cite This Entry:
Radical axis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Radical_axis&oldid=15281
Radical axis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Radical_axis&oldid=15281
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article