Radical axis

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)