Radical axis

The radical axis of two circles $\omega_1, \omega_2$ is defined as the locus of the points $P$ such that the power of $P$ with respect to $\omega_1, \omega_2$ are equal. If $O_i, r_i$ are the center and radius of $\omega_i$ , then we say that for any point $P$ on the radical axis, $PO_1^2 - r_1^2 = PO_2^2 - r_2^2$

The radical axis is a line. If the two circles intersect at two common points, their radical axis is the line through these two points (because the power of each of the common points is $0$ ); if they intersect at one point, their radical axis is the common internal tangent.

The three pairwise radical axes of three circles concur at a point, called the radical center.

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Radical axis