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
\left|\matrix{y_1 & y_2 & y_3\\r_1 & r_2 & r_3\\1 & 1 & 1}\right| y - \left|\matrix{x_1&x_2&x_3\\r_1&r_2&r_3\\1&1&1}\right| x = \left|\matrix{x_1&x_2&x_3\\y_1&y_2&y_3\\r_1&r_2&r_3}\right|
(For some reason, the equation comes out entirely distorted in some browsers. To see the equation you may turn to the backup page.)
References
- G. Salmon, Treatise on Conic Sections, Chelsea Pub, 6e, 1960, p. 107
