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Radical axis

In , the radical axis of two nonconcentric circles is the locus of all points that have equal with respect to both circles, where the of a point P with respect to a circle with O and r is defined as |PO|^2 - r^2. This line can be algebraically derived by subtracting the equations of the two circles, resulting in a that represents the set of such points. Equivalently, it is the set of points from which the lengths of tangents to the two circles are equal. A fundamental property of the radical axis is that it is to the line joining the centers of the two circles. When the circles intersect at two points, the radical axis coincides with the common chord passing through those intersection points. For non-intersecting circles, the radical axis lies between them if one is not contained within the other, and its position along the line of centers can be calculated using the formula d_1 = (d^2 + r_1^2 - r_2^2)/(2d), where d is the distance between centers, and r_1, r_2 are the . The extends to degenerate cases, such as when one "circle" is a point (radius zero), in which the radical axis becomes the bisector or a related line. For three circles, the radical axes of each pair are concurrent at a point known as the radical center, provided the circles are pairwise nonconcentric; this concurrency holds even if the circles do not all intersect. The radical axis plays a crucial role in circle geometry, facilitating theorems on , common tangents, and , and it generalizes to higher dimensions as the radical plane of two spheres.

Definition and Basic Properties

Definition

In , the radical axis of two is defined as the locus of all points that have equal with respect to both . This set of points forms a straight line, provided the circles are not concentric. The of a point P with respect to a circle centered at O with r is given by the formula |PO|^2 - r^2. For two circles with centers O_1, O_2 and radii r_1, r_2, the condition that the powers are equal is |PO_1|^2 - r_1^2 = |PO_2|^2 - r_2^2, which simplifies to the equation of a line to the line joining the centers O_1O_2. To derive the explicit equation in the plane, consider two circles represented in general form as x^2 + y^2 + D_1 x + E_1 y + F_1 = 0 and x^2 + y^2 + D_2 x + E_2 y + F_2 = 0. Subtracting these equations eliminates the quadratic terms, resulting in the linear equation (D_1 - D_2)x + (E_1 - E_2)y + (F_1 - F_2) = 0, which describes the radical axis. The concept of the radical axis, originally discovered by Arab mathematicians, was introduced into modern geometry by in the late 18th century.

Geometric Interpretation

The radical axis of two non-concentric circles is always a straight line consisting of all points that have equal with respect to both circles. This line is to the joining the centers of the two circles. When the two circles intersect at two points, the radical axis coincides with the line containing their common chord. The position of this line relative to the circle centers depends on the radii: if the circles have equal radii, the radical axis is the bisector of the joining the centers; if the radii differ, it is offset along the line of centers, closer to the center of the smaller circle. For non-intersecting circles, the radical axis remains a straight line, which may lie between the circles, externally to both, or in other positions depending on their separation and sizes. Consider two disjoint s, such as one centered at (0,0) with radius 1 and another at (4,0) with radius 1; their radical axis is the vertical line x=2, which separates the plane into regions where the with respect to one circle exceeds that of the other—specifically, points to the left have greater power relative to the left circle, and vice versa.

Fundamental Properties

The radical axis of two circles possesses several key geometric properties that highlight its role as a locus of equal points. One fundamental characteristic is its orientation relative to the circles' centers: the radical axis is always to the line joining the centers of the two circles. This perpendicularity arises from the symmetric nature of the power equality condition and is a direct consequence of the chordal , which describes the locus for points of equal power. The position of the radical axis can be precisely determined relative to each center using the circles' parameters. Let the centers be O_1 and O_2, with radii r_1 and r_2, and let d_c denote the between O_1 and O_2. The signed d from O_1 to the radical axis, measured along the line of centers, is given by d = \frac{r_1^2 - r_2^2 + d_c^2}{2 d_c}. To derive this, consider the circle equations in standard form and subtract them to obtain the radical axis S_1 - S_2 = 0, which simplifies to a . The from O_1 to this line follows from the general formula for the from a point to a line, yielding the above expression after substituting the centers' coordinates and simplifying along the direction. The from O_2 is then d_c - d, ensuring consistency. This formula establishes the axis's location without requiring intersection points, providing essential context for non-intersecting circles. For special configurations, such as two tangent circles, the radical axis coincides with the common tangent line at the point of tangency, where the powers are zero for both circles. In the limiting case of concentric circles (where d_c = 0), the radical axis degenerates to the line at infinity, reflecting the absence of a finite locus due to radially symmetric but differing powers.

Configurations Involving Multiple Circles

Orthogonal Circles

Two circles are said to intersect orthogonally if they cross at right angles, meaning the tangent lines to each circle at their points of intersection are perpendicular. This condition is equivalent to the square of the distance d between their centers equaling the sum of the squares of their radii r_1 and r_2, or d^2 = r_1^2 + r_2^2. At the intersection points, the radii from each center to these points are perpendicular, ensuring the tangents form a 90-degree angle. For two circles that intersect orthogonally, the radical axis is the common chord joining the two intersection points. This line is perpendicular to the line connecting the centers of the circles. The power of each intersection point with respect to both circles is zero, as these points lie on the circles themselves. For any point on this radical axis, the power with respect to both circles is equal, meaning the square of the length of the tangent from that point to one circle equals that to the other; in the orthogonal case, this equal power value determines the radius of circles centered on the axis that intersect both given circles at right angles. Consider the unit circle centered at the origin, with x^2 + y^2 = 1, and another unit circle centered at (\sqrt{2}, 0), with (x - \sqrt{2})^2 + y^2 = 1. The between centers is \sqrt{2}, satisfying (\sqrt{2})^2 = 1^2 + 1^2 = 2. Solving the equations yields intersection points at (\sqrt{2}/2, \sqrt{2}/2) and (\sqrt{2}/2, -\sqrt{2}/2). The radical axis is the vertical line x = \sqrt{2}/2, which is to the horizontal line of centers along the x-axis. For a point P = (\sqrt{2}/2, y) on this axis, the power with respect to both circles is y^2 - 1/2, confirming equality.

Coaxial Circles

A system of circles, also known as a coaxal , is a family of circles such that every pair within the system shares the same radical axis. This fixed radical axis defines the common locus of points with equal with respect to all circles in the system. The system is characterized by two fixed limiting points, which are the points of tangency for the degenerate circles (point circles) in the pencil. Such systems can be generated in two primary ways: the complete coaxial system consists of all circles passing through two fixed distinct points, forming an intersecting pencil where the radical axis is the common chord joining those points; the non-intersecting coaxial system comprises all circles sharing a common radical axis without intersecting at real points, often arising from hyperbolic pencils with the limiting points possibly imaginary. In both cases, the centers of the circles lie on a straight line that is perpendicular to the radical axis and passes through the midpoint of the segment joining the limiting points. This line of centers ensures the geometric coherence of the pencil. The general equation for circles in a system is expressed as a of the equations of two base circles S_1 = 0 and S_2 = 0: \lambda S_1 + \mu S_2 = 0, where \lambda and \mu are parameters (not both zero), yielding the family of circles. The is obtained by setting the coefficient of the terms to zero, corresponding to \lambda / \mu = -a_2 / a_1 in the expanded form. In circle geometry, coaxial systems play a key role in inversion and transformations, where the invariant circles under such transformations often form a pencil with the radical axis serving as a line of or fixed locus. For instance, the circles fixed by a transformation constitute a family, facilitating the analysis of geometric mappings in the .

Systems of Orthogonal Circles

A system of orthogonal circles refers to a of circles in which every member intersects orthogonally with every circle in a given . Such systems arise naturally as conjugate pairs in the classification of circle pencils, where one is orthogonal to its conjugate counterpart. In , these orthogonal s play a fundamental role, as inversion transformations preserve angles and thus map orthogonal circles to orthogonal circles. Specifically, given a —characterized by all members sharing a common radical axis—the orthogonal consists of all circles that intersect each member of the at right angles. This duality ensures that pencils come in orthogonal pairs: an elliptic (circles through two fixed points) is orthogonal to a (Apollonian-type circles with collinear centers), and a parabolic (concentric circles) is orthogonal to another parabolic or degenerate cases involving lines. Since the orthogonal pencil is itself coaxial, all pairs of its circles share a common radical axis, distinct from that of the original pencil and tied to the conjugate coaxial structure, reflecting the shared geometric invariants under inversion. A representative example involves circles orthogonal to a fixed circle C with center O and radius R. For a circle with center I and radius r orthogonal to C, the condition |OI|^2 = R^2 + r^2 holds, and the radical axis of this circle with C is the polar line of I with respect to C. This polar line serves as the locus of points with equal power relative to both circles, illustrating how orthogonality links radical axes to pole-polar relations in the fixed circle. In the parabolic case, where the coaxial consists of concentric s centered at a point O, the orthogonal system degenerates to the pencil of all straight lines passing through O, which intersect every concentric at right angles.)

Radical Center

For Three Circles

The center of three s is defined as the unique point that possesses equal with respect to each of the three s. This point serves as the of the axes formed by any two pairs of the circles, and the radical axis of the third pair concurs at the same location due to the concurrency property inherent in circle geometry. The concurrency of the radical axes for three circles is guaranteed by the radical center theorem, which states that the pairwise radical axes either intersect at a single point or are all parallel, provided no two circles are concentric, ensuring the axes meet unless the centers of the circles are collinear, in which case the axes are parallel and the radical center lies at . If the three circles pass through a common point, that point coincides with the radical center, as with respect to each circle is zero there. A key property of the radical center is that the common power value k at this point with respect to all three circles determines a unique circle orthogonal to each of them, known as the common radical circle or coaxial circle for the system; specifically, a circle centered at the radical center with radius \sqrt{-k} (when k < 0) intersects each of the three circles orthogonally. In cases where the radical axes are parallel, no finite real radical center exists, and the circles belong to a coaxial family sharing a common radical axis at infinity. Configurations yielding an imaginary radical center arise in complex plane extensions but are not realized in the real plane beyond the parallel case. For an illustrative example, consider three mutually tangent circles, such as those externally tangent to one another. The radical center in this setup is the point from which the lengths of the tangent segments to each circle are equal, analogous to the incenter of a triangle being equidistant from its sides; this point facilitates constructions like the Apollonius circles tangent to all three.

Construction Methods

The algebraic construction of the radical axis for two circles begins with their standard equations in coordinate form. Consider two circles given by the general equations S_1 = x^2 + y^2 + D_1 x + E_1 y + F_1 = 0 and S_2 = x^2 + y^2 + D_2 x + E_2 y + F_2 = 0. Subtracting these equations eliminates the quadratic terms, yielding the linear equation S_1 - S_2 = (D_1 - D_2)x + (E_1 - E_2)y + (F_1 - F_2) = 0, which represents the as a straight line. This method is efficient for computational purposes and extends readily to finding intersections with other lines or curves. For geometric constructions using compass and straightedge, the approach varies based on the relative positions of the circles. If the circles intersect at two points, the radical axis is simply the common chord, constructed by joining those intersection points directly. For non-intersecting circles (separate or one inside the other without touching), one standard method involves introducing a third auxiliary circle that intersects both given circles at two points each. Construct the common chords of the auxiliary circle with each of the given circles; these chords intersect at a point on the radical axis. The radical axis is then the line passing through this intersection point and perpendicular to the line joining the centers of the two original circles. Alternatively, for separate circles, draw the four common tangents (two external and two internal); the midpoints of these tangent segments lie on the radical axis, allowing it to be constructed by connecting any two such midpoints. If the circles touch externally or internally, the radical axis coincides with the common tangent at the point of contact. To construct the radical center of three circles, first find the radical axes of two pairs (e.g., circles 1 and 2, and circles 1 and 3) using the methods above; their intersection point is the radical center, which automatically lies on the third radical axis due to concurrency. Verification with the third axis confirms the construction, ensuring accuracy in practical applications like locating points of equal power. In projective geometry, the radical axis can also be constructed using poles and polars with respect to one of the circles. The common external tangents to the two circles intersect at the external center of similitude, and the common internal tangents intersect at the internal center of similitude. The radical axis is the polar line of either of these similitude centers with respect to one of the circles, obtained by constructing the polar via the circle's inversion properties or harmonic divisions. Historically, Gaspard Monge contributed to the foundational understanding of the radical axis in the late 18th century through his work on circle systems and orthogonal trajectories, where constructions often involved perpendiculars to lines of centers to locate axes in descriptive geometry settings.

Coordinate Systems and Representations

Bipolar Coordinates

Bipolar coordinates constitute a two-dimensional orthogonal curvilinear system defined relative to two foci, conventionally located at (-a, 0) and (a, 0) in the Cartesian plane, where a > 0. The coordinates (\tau, \sigma) of a point P(x, y) are determined by the distances r_1 and r_2 from P to the respective foci, with \tau = \ln\left(\frac{r_1}{r_2}\right) and \sigma representing the angle subtended at P by the segment joining the foci. These coordinates transform the plane such that the metric scale factors facilitate solving partial differential equations in regions bounded by circular arcs. The level curves of constant \tau form a family of coaxial circles centered along the x-axis (the line of foci), while constant \sigma curves form another coaxial family centered along the y-axis (the perpendicular bisector of the foci). Each family shares a common radical axis: for the \tau = constant circles, it is the y-axis (\tau = 0); for the \sigma = constant circles, it is the x-axis (\sigma = \pi/2 or equivalent degenerate line). Consequently, the radical axis of any two circles from the \tau = constant family manifests as a line of constant \sigma, simplifying the geometric analysis of power equality across such pairs. This embedding of coaxial circle systems into the coordinate framework highlights the radical axis as an intrinsic coordinate line, with the perpendicular bisector serving as the shared radical axis for non-intersecting coaxial circles in the \tau family. In applications, bipolar coordinates prove particularly useful in and , where circular boundaries arise, such as in the configuration of two cylindrical conductors forming a . The \tau coordinate corresponds directly to the logarithmic potential generated by equal and opposite line charges at the foci, enabling separable solutions to in these coordinates: \nabla^2 \phi = 0 yields \phi(\tau, \sigma) = \tau f(\sigma) + g(\sigma), with boundary conditions on constant-\tau cylinders yielding explicit capacitance formulas like C = \frac{2\pi \epsilon_0}{|\tau_2 - \tau_1|}. This relation to Cartesian coordinates via the logarithmic potential \phi \propto \ln(r_1 / r_2) underscores the system's utility for modeling fields in multiply connected domains with circular geometry.

Trilinear Coordinates

Trilinear coordinates provide a homogeneous system for locating points in the plane of a triangle \triangle ABC, where the coordinates (x : y : z) of a point P are proportional to the signed distances from P to the sides BC, CA, and AB, respectively. These coordinates are barycentric-like but normalized by the side lengths, offering a natural framework for expressing geometric objects relative to the triangle's sides. In this , the equation of a takes the specific form (l x + m y + n z)(x + y + z) = x^2 + y^2 + z^2, where l, m, n are constants determined by the 's position and size relative to \triangle [ABC](/page/ABC). This representation highlights the 's linear variation across the . The radical axis of two such s, with parameters (l_1, m_1, n_1) and (l_2, m_2, n_2), is obtained by subtracting their equations, yielding the (l_1 - l_2) x + (m_1 - m_2) y + (n_1 - n_2) z = 0, which defines a straight line as expected. For the radical center of three circles with linear forms L_1 = l_1 x + m_1 y + n_1 z, L_2 = l_2 x + m_2 y + n_2 z, and L_3 = l_3 x + m_3 y + n_3 z, the center is the point where the powers with respect to all are equal. This occurs where L_1 = L_2 = L_3, or equivalently, at the intersection of the radical axes L_1 - L_2 = 0 and L_1 - L_3 = 0. The (x : y : z) solve this homogeneous and can be expressed using determinants: x : y : z = \det \begin{pmatrix} m_1 - m_2 & n_1 - n_2 \\ m_1 - m_3 & n_1 - n_3 \end{pmatrix} : -\det \begin{pmatrix} l_1 - l_2 & n_1 - n_2 \\ l_1 - l_3 & n_1 - n_3 \end{pmatrix} : \det \begin{pmatrix} l_1 - l_2 & m_1 - m_2 \\ l_1 - l_3 & m_1 - m_3 \end{pmatrix}. This yields the explicit location in terms of the circles' parameters. In triangle geometry, this representation facilitates identifying radical centers of notable circle systems. For instance, consider the of \triangle [ABC](/page/ABC) and the three circles having the sides BC, CA, and AB as diameters. The radical axes of the with each diameter circle are the altitudes of the , so their radical center is the orthocenter, with \cos A : \cos B : \cos C. More generally, for any three cevians through different vertices, the radical center of the circles with those cevians as diameters is the orthocenter of \triangle [ABC](/page/ABC). If the cevians are the altitudes themselves, the resulting circles (with diameters along the altitudes) also have their radical center at the orthocenter.

Generalizations to Higher Dimensions

Radical Plane

The radical plane of two spheres in three-dimensional space generalizes the concept of the radical axis from the , serving as the locus of all points that have equal power with respect to both spheres. The power of a point with respect to a sphere is the value obtained by substituting the point's coordinates into the sphere's (normalized so the quadratic terms have 1), or equivalently, the product of the directed distances from the point to the intersection points of any line through it with the sphere. This locus forms a , and any point on it satisfies the condition that the lengths of tangents from the point to each sphere are equal. The equation of the radical plane can be derived by setting the powers equal, which corresponds to subtracting the equations of the two spheres. For two spheres given by S_1: x^2 + y^2 + z^2 + D_1 x + E_1 y + F_1 z + G_1 = 0 and S_2: x^2 + y^2 + z^2 + D_2 x + E_2 y + F_2 z + G_2 = 0, the radical plane is S_1 - S_2 = 0 \quad \Leftrightarrow \quad (D_1 - D_2)x + (E_1 - E_2)y + (F_1 - F_2)z + (G_1 - G_2) = 0. Equivalently, in center-radius form, for spheres with centers (a_1, b_1, c_1) and (a_2, b_2, c_2), and radii r_1 and r_2, the equation simplifies to $2(a_2 - a_1)x + 2(b_2 - b_1)y + 2(c_2 - c_1)z = (a_1^2 + b_1^2 + c_1^2 - r_1^2) - (a_2^2 + b_2^2 + c_2^2 - r_2^2). This confirms that the locus is indeed a . Key properties of the radical plane include its perpendicularity to the line joining the centers of the two s. The radical plane intersects this line at a \frac{d^2 + r_1^2 - r_2^2}{2d} from the center of the first , where d is the between centers. If the two s intersect, their intersection forms a circle lying entirely within the radical plane. The radical plane remains invariant for any pair of spheres in a system, where all spheres share the same line of centers. In special cases, when the two spheres have equal radii, the radical plane is the perpendicular bisecting plane midway between the centers. If the spheres are concentric (sharing the same center but different radii), no real radical plane exists, as the difference of their equations yields a constant rather than a linear form, resulting in either the entire space or no points satisfying the equal-power condition unless the spheres coincide. For tangent spheres, the radical plane passes through the point of tangency.

Radical Hyperplane

In n-dimensional , the radical hyperplane of two hyperspheres is defined as the (n-1)-dimensional affine subspace consisting of all points that have equal with respect to both hyperspheres. The of a point \mathbf{x} with respect to a hypersphere centered at \mathbf{c} with radius r is given by \|\mathbf{x} - \mathbf{c}\|^2 - r^2. This locus generalizes the radical axis from 2 dimensions and the radical plane from 3 dimensions to arbitrary dimensions. For two hyperspheres S_1: \|\mathbf{x} - \mathbf{c_1}\|^2 = r_1^2 and S_2: \|\mathbf{x} - \mathbf{c_2}\|^2 = r_2^2, setting the powers equal yields the equation of the radical hyperplane: $2(\mathbf{c_2} - \mathbf{c_1}) \cdot \left( \mathbf{x} - \frac{\mathbf{c_1} + \mathbf{c_2}}{2} \right) = r_2^2 - r_1^2. This describes a perpendicular to the line joining the centers \mathbf{c_1} and \mathbf{c_2}. The hyperplane always exists and is unique for non-concentric hyperspheres, as the difference in their defining equations degenerates to a . For a collection of hyperspheres, the is the point of their pairwise hyperplanes, provided such a point exists, and it is the unique point equidistant in from all hyperspheres in the . Applications of radical hyperplanes appear in higher-dimensional , such as defining the boundaries in additively weighted Voronoi diagrams for hyperspheres, where cells consist of points closer in to one site than others. In , the concept extends to general hypersurfaces in n dimensions, facilitating the analysis of pencils of with identical quadratic terms, whose difference yields a as the common .