Radical Axis of Two Circles

The power of a point $P=(x_1,y_1)$ with respect to a circle $S\equiv x^2+y^2+2gx+2fy+c=0$ is the number $S_1=x_1^2+y_1^2+2gx_1+2fy_1+c$ obtained by substituting $P$'s coordinates into the equation's left side; when $P$ lies outside the circle, $S_1$ equals the square of the length of the tangent from $P$ to the circle.

The radical axis of two circles $S=0$ and $S'=0$ is the locus of points whose power is the same with respect to both -- equivalently, the points from which the two circles have equal tangent length. Setting the two powers equal, $S=S'$, i.e.

and noting that the $x^2,y^2$ terms cancel (both circles have leading coefficient $1$), shows the radical axis is always a genuine straight line:

whatever the relative position of the two circles -- even circles that do not meet in real points still have a real radical axis. When the circles do intersect, this line coincides with their common chord (or common tangent, if they merely touch), since a genuine intersection point trivially has equal, zero-deficit power with respect to both equations. …