Radical axis

In geometry, the radical axis of two non-concentric circles is a line defined from the two circles, perpendicular to the line connecting the centers of the circles. If the circles cross, their radical axis is the line through their two crossing points, and if they are tangent, it is their line of tangency. For two disjoint circles, the radical axis is the locus of points at which tangents drawn to both circles have equal lengths.

Figure 1. Illustration of the radical axis (red line) of two given circles (solid black). For any point P (blue) on the radical axis, a unique circle (dashed) can be drawn that is centered on that point and intersects both given circles at right angles, i.e., orthogonally, at the points where the tangents from P touch the circles. The point P has an equal power with respect to both given circles, because the tangents from P (blue lines) are radii of the orthogonal circle and thus have equal length.

Regardless of whether the two defining circles cross, are tangent, or are disjoint, their radical axis is the locus of points whose power with respect to the two circles is equal. For this reason the radical axis is also called the power line or power bisector of the two circles. The power of a point with respect to a circle is the squared Euclidean distance from the point to the circle's center, minus the squared radius of the circle. For a point $p$ outside a circle $C$ , its power is a positive number, the radius of another circle centered at $p$ that crosses $C$ at right angles. Therefore, the points of the radical axis that are outside of its defining circles are the centers of circles that cross both defining circles at right angles.[1]

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.

If the two circles intersect, the radical axis is the secant line corresponding to their common chord.

Figure 2: The radical center (orange point) is the center of the unique circle (also orange) that intersects three given circles at right angles.

Radical center of three circles

Consider three circles A, B and C, no two of which are concentric. The radical axis theorem states that the three radical axes (for each pair of circles) intersect in one point called the radical center, or are parallel.[2] In technical language, the three radical axes are concurrent (share a common point); if they are parallel, they concur at a point of infinity.

A simple proof is as follows.[3] The radical axis of circles A and B is defined as the line along which the tangents to those circles are equal in length a=b. Similarly, the tangents to circles B and C must be equal in length on their radical axis. By the transitivity of equality, all three tangents are equal a=b=c at the intersection point r of those two radical axes. Hence, the radical axis for circles A and C must pass through the same point r, since a=c there. This common intersection point r is the radical center.

There is a unique circle with its center at the radical center that is orthogonal to all three circles. This follows, also by transitivity, because each radical axis, being the locus of centers of circles that cut each pair of given circles orthogonally, requires all three circles to have equal radius at the intersection of all three axes.

Geometric construction

Radical Axis Construction

The radical axis of two circles A and B can be constructed by drawing a line through any two points on the axis. A point on the axis can be found by drawing a circle C that intersects both circles A and B in two points. The two lines passing through each pair of intersection points are the radical axes of A and C and of B and C. These two lines intersect in a point J that is the radical center of all three circles, as described above; therefore, this point also lies on the radical axis of A and B. Repeating this process with another such circle D provides a second point K. The radical axis is the line passing through both J and K.

Figure 3: Lines through corresponding antihomologous points intersect on the radical axis of the two given circles (green and blue, centered on points C and D, respectively). The points P and Q are antihomologous, as are S and T. These four points lie on a circle that intersects the two given circles.

A special case of this approach, seen in Figure 3, is carried out with antihomologous points from an internal or external center of similarity. Consider two rays emanating from an external homothetic center E. Let the antihomologous pairs of intersection points of these rays with the two given circles be denoted as P and Q, and S and T, respectively. These four points lie on a common circle that intersects the two given circles in two points each.[4] Hence, the two lines joining P and S, and Q and T intersect at the radical center of the three circles, which lies on the radical axis of the given circles.[5] Similarly, the line joining two antihomologous points on separate circles and their tangents form an isosceles triangle, with both tangents being of equal length.[6] Therefore, such tangents meet on the radical axis.[5]

Algebraic construction

Figure 4: Finding the radical axis algebraically. L is the distance from J to K, whereas x₁ and x₂ are the distances from K to B and from K to V, respectively. The variables d₁ and d₂ represent the distances from J to B and from J to V, respectively.

Referring to Figure 4, the radical axis (red) is perpendicular to the blue line segment joining the centers B and V of the two given circles, intersecting that line segment at a point K between the two circles. Therefore, it suffices to find the distance x₁ or x₂ from K to B or V, respectively, where x₁+x₂ equals D, the distance between B and V.

Consider a point J on the radical axis, and let its distances to B and V be denoted as d₁ and d₂, respectively. Since J must have the same power with respect to both circles, it follows that

d_{1}^{2}-r_{1}^{2}=d_{2}^{2}-r_{2}^{2}

where r₁ and r₂ are the radii of the two given circles. By the Pythagorean theorem, the distances d₁ and d₂ can be expressed in terms of x₁, x₂ and L, the distance from J to K

L^{2}+x_{1}^{2}-r_{1}^{2}=L^{2}+x_{2}^{2}-r_{2}^{2}

By cancelling L² on both sides of the equation, the equation can be written

x_{1}^{2}-x_{2}^{2}=r_{1}^{2}-r_{2}^{2}

Dividing both sides by D = x₁+x₂ yields the equation

x_{1}-x_{2}={\frac {r_{1}^{2}-r_{2}^{2}}{D}}

Adding this equation to x₁+x₂ = D yields a formula for x₁

2x_{1}=D+{\frac {r_{1}^{2}-r_{2}^{2}}{D}}

Subtracting the same equation yields the corresponding formula for x₂

2x_{2}=D-{\frac {r_{1}^{2}-r_{2}^{2}}{D}}

Determinant calculation

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:

(dx + ey + fz)(ax + by + cz) + g(ayz + bzx + cxy) = 0

(hx + iy + jz)(ax + by + cz) + k(ayz + bzx + cxy) = 0

(lx + my + nz)(ax + by + cz) + p(ayz + bzx + cxy) = 0

Then the radical center is the point

\det {\begin{bmatrix}g&k&p\\e&i&m\\f&j&n\end{bmatrix}}:\det {\begin{bmatrix}g&k&p\\f&j&n\\d&h&l\end{bmatrix}}:\det {\begin{bmatrix}g&k&p\\d&h&l\\e&i&m\end{bmatrix}}.

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.[7] 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.

Notes

Johnson (1960), pp. 31–32.
Johnson (1960), pp. 32–33.
Johnson (1960), p. 32.
Johnson (1960), pp. 20–21.
Johnson (1960), p. 41.
Johnson (1960), p. 21.
See Merriam–Webster online dictionary.

References

R. A. Johnson (1960). Advanced Euclidean Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle (reprint of 1929 edition by Houghton Miflin ed.). New York: Dover Publications. pp. 31–43. ISBN 978-0-486-46237-0.

External links

Weisstein, Eric W. "Radical line". MathWorld.
Weisstein, Eric W. "Chordal theorem". MathWorld.
Animation at Cut-the-knot

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[1] Johnson (1960), pp. 31–32.

[2] Johnson (1960), pp. 32–33.

[3] Johnson (1960), p. 32.

[4] Johnson (1960), pp. 20–21.

[Johnson_1960,_p._41-5] Johnson (1960), p. 41.

[6] Johnson (1960), p. 21.

[7] See Merriam–Webster online dictionary.