Radical axis

In Euclidean 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 with centers M_1, M_2 and radii r_1, r_2 the powers of a point P with respect to the circles are Point 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 is outside the circles, 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. In any case the radical axis is a line perpendicular to On notations The notation radical axis was used by the French mathematician M. Chasles as axe radical. J.V. Poncelet used chorde ideale. J. Plücker introduced the term Chordale. J. Steiner called the radical axis line of equal powers (Linie der gleichen Potenzen) which led to power line (Potenzgerade). Let be the position vectors of the points . Then the defining equation of the radical line can be written as: From the right equation one gets The pointset of the radical axis is indeed a line and is perpendicular to the line through the circle centers. ( is a normal vector to the radical axis !) Dividing the equation by , one gets the Hessian normal form. Inserting the position vectors of the centers yields the distances of the centers to the radical axis: with . ( may be negative if is not between .) If the circles are intersecting at two points, the radical line runs through the common points. If they only touch each other, the radical line is the common tangent line. The radical axis of two intersecting circles is their common secant line. The radical axis of two touching circles is their common tangent. The radical axis of two non intersecting circles is the common secant of two convenient equipower circles (see below). For a point outside a circle and the two tangent points the equation holds and lie on the circle with center and radius .