Radical axis

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The radical axis of two circles is the locus of points at which tangents drawn to both circles have the same length; in technical language, all points on the radical axis have the same power with respect to both circles.[1]

The radical axis is always a straight line and always perpendicular to the line connecting the centers of the circles, albeit closer to the circumference of the larger circle. If the circles intersect, the radical axis is the line passing through the intersection points; similarly, if the circles are tangent, the radical axis is simply the common tangent. In general, two disjoint, non-concentric circles can be aligned with the circles of bipolar coordinates; in that case, the radical axis is simply the y-axis; every circle on that axis that passes through the two foci intersect the two circles orthogonally. Thus, two radii of such a circle are tangent to both circles, satisfying the definition of the radical axis.

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[edit] Constructing the radical axis for two circles

Being a line, the radical axis of two circles A and B can be found by drawing a line through two of its points. To find one such point, draw a circle C that intersects both circles A and B in two points, a1 and a2, and b1 and b2, respectively. Since all four intersection points all lie on the same circle C, the secant chords have the same power; denoting the intersection point of these chords as p1

\mathbf{p}_{1}\mathbf{a}_{1} \cdot \mathbf{p}_{1}\mathbf{a}_{2} = \mathbf{p}_{1}\mathbf{b}_{1} \cdot \mathbf{p}_{1}\mathbf{b}_{2}

Therefore, p1 lies on the radical axis. Repeating the construction for a different circle D that again intersects both A and B in two points provides a second point p2. The radical axis is the line passing through both p1 and p2.

[edit] Radical center

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.

[edit] Relationship with homothetic centers

Consider two circles A and B, and on them, let there be two pairs of antihomologous points a1 and a2 being antihomologous to b1 and b2, respectively. Then all four points lie on a common circle C,[4] and by the construction above, the secant lines a1 a2 and b1 b2 intersect on the radical axis.[5]

Similarly, the line joining two antihomologous points on separate circles and their tangents form an isoceles triangle, with both tangents being of equal length.[6] Therefore, such tangents meet on the radical axis.[7]

[edit] 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

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

[edit] References

  1. ^ Johnson (1960), pp. 31–32.
  2. ^ Johnson (1960), pp. 32–33.
  3. ^ Johnson (1960), p. 32.
  4. ^ Johnson (1960), pp. 20–21.
  5. ^ Johnson (1960), p. 41.
  6. ^ Johnson (1960), p. 21.
  7. ^ Johnson (1960), p. 41.
  • Roger A. Johnson, Modern Geometry, Houghton Mifflin, Cambridge, Massachusetts, 1929. Dover reprint, 1960.
  • Clark Kimberling, "Triangle Centers and Central Triangles," Congressus Numerantium 129 (1998) i-xxv, 1-295.