Radical axis
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 formula_1 with centers formula_2 and radii formula_3 the powers of a point formula_4 with respect to the circles are Point formula_4 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 formula_4 is  outside the circles, formula_4 has equal tangential distance to both the circles. If the radii are equal, the radical axis is the line segment bisector of formula_2. In any case the radical axis is a line perpendicular to formula_11. 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). Properties. Geometric shape and its position. Let formula_12 be the position vectors of the points  formula_13. Then the defining equation of th radical line can be written as: From the right equation one gets (formula_15 is a normal vector to the radical axis !) Dividing the equation by formula_16, one gets the Hessian normal form. Inserting the position vectors of the centers yields the distances of the centers to the radical axis: 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. 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 formula_1 be two apart lying circles (as in the previous section), formula_37 their centers and radii and formula_38 their radical axis. Now, all circles will be determined with centers on line formula_11, which have together with formula_40 line formula_38 as radical axis, too. If formula_42 is such a circle, whose center has distance formula_43 to the center formula_44 and radius formula_45. From the result in the previous section one gets the equation With formula_48 the  equation can be rewritten as: If radius formula_45 is given, from this equation one finds the distance formula_51 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 formula_1 orthogonally and hence all new circles (purple), too. Choosing the (red) radical axis as y-axis and line formula_11 as x-axis, the two pencils of circles have the equations: Properties: a) Any two green circles intersect on the x-axis at the points formula_57 , 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 formula_58. Special cases: a) In case of formula_59 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 formula_40 to its center formula_44, i. e. formula_62, the equations turn into a more simple form and one gets formula_63. Conclusion: a) For any real formula_64 the pencil of circles b) For any real formula_64 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 formula_77: In case of formula_106 the axes have to be chosen additionally. The system is parabolic and can be drawn easily. Coaxal circles . Definition and properties: Let formula_1 be two circles and formula_108 their power functions. Then for any formula_109 is the equation of a circle formula_111 (see below). Such a system of circles is called coaxal circles generated by the circles formula_1. (In case of formula_113 the equation describes the radical axis of formula_1.) The power function of formula_111 is The "normed" equation (the coefficients of formula_117 are formula_118) of formula_111 is formula_120. A simple calculation shows: Allowing formula_123 to move to infinity, one recognizes, that formula_1 are members of the system of coaxal circles: formula_125. (E): If formula_1 "intersect" at two  points formula_77, any circle formula_111 contains formula_77, too, and line formula_83 is their common radical axis. Such a system is called "elliptic". (P): If formula_1 are "tangent" at formula_4, any circle is tangent to formula_1 at point formula_4, too. The common tangent is their common radical axis. Such a system is called "parabolic". (H): If formula_1 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 formula_1. 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 formula_139 gives the normed circle equation: Completing the square and the substitution formula_141 (x-coordinate of the center) yields the centered form of the equation In case of formula_143 the circles formula_144 have the two points in common and the system of coaxal circles is "elliptic". In case of formula_146 the circles formula_144 have point formula_148 in common and the system is "parabolic". In case of formula_149 the circles formula_144 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 infinite large radius. For example: describes all circles, which have with the first circle the line formula_153 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 formula_155 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 formula_175, lie on a circle concentric to formula_175. Let call it "equipower circle". This property can be used for an additional construction method of the radical axis of two circles: For two non intersecting circles formula_1 there can be drawn two equipower circles formula_178, which have the  same power with respect to formula_1 (see diagram). In detail: formula_180. If the power is large enough, the circles formula_178 have two points in common, which lie on the radical axis formula_38. 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 formula_183-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.