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In geometry, the radical axis of two non-concentric circles is the set of points whose power with respect to the circles are equal. For this reason the radical axis is also called the power line or power bisector of the two circles. In detail:
For two circles c 1 , c 2 {\displaystyle c_{1},c_{2}} with centers M 1 , M 2 {\displaystyle M_{1},M_{2}} and radii r 1 , r 2 {\displaystyle r_{1},r_{2}} the powers of a point P {\displaystyle P} with respect to the circles are
Point P {\displaystyle P} belongs to the radical axis, if
If the circles have two points in common, the radical axis is the common secant line of the circles. If point P {\displaystyle P} is outside the circles, P {\displaystyle P} has equal tangential distance to both the circles.If the radii are equal, the radical axis is the line segment bisector of M 1 , M 2 {\displaystyle M_{1},M_{2}}.In any case the radical axis is a line perpendicular to M 1 M 2 ¯ {\displaystyle {\overline {M_{1}M_{2}}}}.