[[File:Potenz-gerade-def.svg|thumb|upright=1.4| The tangent lines must be equal in length for any point on the radical axis: If P, T1, T2 lie on a common tangent, then P is the midpoint of \overline{T_1T_2}.]] 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 c1, c2 with centers M1, M2 and radii r1, r2 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 M1, M2 . In any case the radical axis is a line perpendicular to 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 which led to power line (Potenzgerade).

Properties

Geometric shape and its position

Let be the position vectors of the points P,M_1,M_2. Then the defining equation of the radical line can be written as: From the right equation one gets ( 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: (d_i may be negative if L is not between M_1,M_2.) 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.

Special positions

Orthogonal circles

System of orthogonal circles

The method described in the previous section for the construction of a pencil of circles, which intersect two given circles orthogonally, can be extended to the construction of two orthogonally intersecting systems of circles: Let c_1,c_2 be two apart lying circles (as in the previous section), their centers and radii and g_{12} their radical axis. Now, all circles will be determined with centers on line , which have together with c_1 line g_{12} as radical axis, too. If \gamma_2 is such a circle, whose center has distance \delta to the center M_1 and radius \rho_2. From the result in the previous section one gets the equation With the equation can be rewritten as: If radius \rho_2 is given, from this equation one finds the distance \delta_2 to the (fixed) radical axis of the new center. In the diagram the color of the new circles is purple. Any green circle (see diagram) has its center on the radical axis and intersects the circles c_1,c_2 orthogonally and hence all new circles (purple), too. Choosing the (red) radical axis as y-axis and line as x-axis, the two pencils of circles have the equations: (; (0,y_g) is the center of a green circle.) Properties: a) Any two green circles intersect on the x-axis at the points, the poles of the orthogonal system of circles. That means, the x-axis is the radical line of the green circles. b) The purple circles have no points in common. But, if one considers the real plane as part of the complex plane, then any two purple circles intersect on the y-axis (their common radical axis) at the points. Special cases: a) In case of d_1=r_1 the green circles are touching each other at the origin with the x-axis as common tangent and the purple circles have the y-axis as common tangent. Such a system of circles is called coaxal parabolic circles (see below). b) Shrinking c_1 to its center M_1, i. e. r_1=0, the equations turn into a more simple form and one gets M_1=P_1. Conclusion: a) For any real w the pencil of circles b) For any real w the two pencils of circles c) From the equations in b), one gets a coordinate free representation: Straightedge and compass construction: A system of orthogonal circles is determined uniquely by its poles P_1,P_2: In case of P_1=P_2 the axes have to be chosen additionally. The system is parabolic and can be drawn easily.

Coaxal circles

Definition and properties: Let c_1,c_2 be two circles and \Pi_1,\Pi_2 their power functions. Then for any is the equation of a circle c(\lambda) (see below). Such a system of circles is called coaxal circles generated by the circles c_1,c_2. (In case of \lambda=1 the equation describes the radical axis of c_1,c_2.) The power function of c(\lambda) is The normed equation (the coefficients of x^2,y^2 are 1) of c(\lambda) is. A simple calculation shows: Allowing \lambda to move to infinity, one recognizes, that c_1,c_2 are members of the system of coaxal circles:. (E): If c_1,c_2 intersect at two points P_1,P_2, any circle c(\lambda) contains P_1,P_2, too, and line is their common radical axis. Such a system is called elliptic. (P): If c_1,c_2 are tangent at P, any circle is tangent to c_1,c_2 at point P, too. The common tangent is their common radical axis. Such a system is called parabolic. (H): If c_1,c_2 have no point in common, then any pair of the system, too. The radical axis of any pair of circles is the radical axis of c_1,c_2. The system is called hyperbolic. In detail: Introducing coordinates such that then the y-axis is their radical axis (see above). Calculating the power function gives the normed circle equation: Completing the square and the substitution (x-coordinate of the center) yields the centered form of the equation In case of r_1>d_1 the circles have the two points in common and the system of coaxal circles is elliptic. In case of r_1=d_1 the circles have point P_0=(0,0) in common and the system is parabolic. In case of r_1<d_1 the circles have no point in common and the system is hyperbolic. Alternative equations: 1) In the defining equation of a coaxal system of circles there can be used multiples of the power functions, too. 2) The equation of one of the circles can be replaced by the equation of the desired radical axis. The radical axis can be seen as a circle with an infinitely large radius. For example: describes all circles, which have with the first circle the line x=x_2 as radical axis. 3) In order to express the equal status of the two circles, the following form is often used: But in this case the representation of a circle by the parameters \mu,\nu is not unique. Applications: a) Circle inversions and Möbius transformations preserve angles and generalized circles. Hence orthogonal systems of circles play an essential role with investigations on these mappings. b) In electromagnetism coaxal circles appear as field lines.

Radical center of three circles, construction of the radical axis

Additional construction method: All points which have the same power to a given circle c lie on a circle concentric to c. Let us call it an equipower circle. This property can be used for an additional construction method of the radical axis of two circles: For two non intersecting circles c_1,c_2, there can be drawn two equipower circles c'_1,c'_2, which have the same power with respect to c_1,c_2 (see diagram). In detail:. If the power is large enough, the circles c'_1,c'2 have two points in common, which lie on the radical axis g{12}.

[Radical center of three circles

The green circle intersects the three circles orthogonally. | upload.wikimedia.org/wikipedia/commons/5/5f/Potenz-gerade-3k.svg]

Relation to bipolar coordinates

In general, any two disjoint, non-concentric circles can be aligned with the circles of a system of bipolar coordinates. In that case, the radical axis is simply the y-axis of this system of coordinates. Every circle on the axis that passes through the two foci of the coordinate system intersects the two circles orthogonally. A maximal collection of circles, all having centers on a given line and all pairs having the same radical axis, is known as a pencil of coaxal circles.

Radical center in trilinear coordinates

If the circles are represented in trilinear coordinates in the usual way, then their radical center is conveniently given as a certain determinant. Specifically, let X = x : y : z denote a variable point in the plane of a triangle ABC with sidelengths a = |BC|, b = |CA|, c = |AB|, and represent the circles as follows: Then the radical center is the point

Radical plane and hyperplane

The radical plane of two nonconcentric spheres in three dimensions is defined similarly: it is the locus of points from which tangents to the two spheres have the same length. The fact that this locus is a plane follows by rotation in the third dimension from the fact that the radical axis is a straight line. The same definition can be applied to hyperspheres in Euclidean space of any dimension, giving the radical hyperplane of two nonconcentric hyperspheres.