Any two circles, if not congruent, are necessarily similar. When the circles do not intersect and lie outside each other, they have two centers of similitude. One is found at the intersection of the common external tangents of the two circles, the other at the intersection of their internal common tangents.
We'll denote the circle with center at (x, y) and radius r as C(x, y, r) . Let there be two circles, C(x_1, y_1, r_1) and C(x_2, y_2, r_2) .
The internal center of similitude divides the segment joining their centers in the ratio of their radii:
S_i = \Big(\frac{x_2 r_1 + x_1 r_2}{r_1 + r_2}, \frac{y_2 r_1 + y_1 r_2}{r_1 + r_2}\Big).

The same is true of the external center. The division is external, though:
S_e = \Big(\frac{x_2 r_1  x_1 r_2}{r_1  r_2}, \frac{y_2 r_1  y_1 r_2}{r_1  r_2}\Big).

If it were not for the difference in signs, the formulas would be practically the same. Both centers (when exist) lie on the center line of the two circles.