Axis of Similitude of Three Circles

Given three non-intersecting circles C(x_i, y_i, r_i), i = 1, 2, 3, that lie outside each other and have different radii, their centers of similitude are collinear, i.e., lie on a single straight line, known as the axis of similitude (or Monge Line) of the three circles. The line has the following equation

Ax - By = C,

where

A =

y1 y2 y3 r1 r2 r3 1 1 1

,

B =

x1 x2 x3 r1 r2 r3 1 1 1

,

C =

x1 x2 x3 y1 y2 y3 r1 r2 r3

References

1. G. Salmon, Treatise on Conic Sections, Chelsea Pub, 6e, 1960, p. 107

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