In general any two circles have two centers of homothety. They have only one center when the circles have the same radius or when the circles have the same center.
Given two circles of different radii with disjoint interiors, their external center of homothety is at the intersection of the external tangents and the internal center is at the intersection of the internal tangents. This is the case because homothety preserves tangency. So the centers of homothety are easy to construct in this case.
If the circles partially overlap, there is still an external homothetyic center at the intersection of external tangents, and an internal homothety center lying somewhere along the line joining the centers of the circles. If the circles overlap completlety (one inside the other), there are two internal homothety centers on the line joining the centers of the circles.
My question is how do you construct (with compass and straightedge) the centers of homothety in the cases where the circle interiors are not disjoint.
Best Answer
An easy property you can use to construct the center of homothety is this: an homothety will preserve the angle of a point along the circle (measured from the line joining the center of homothety and the center of the circle). These pictures make it pretty clear (links to imgur).
Direct homothety.
If the homothety is inverse (meaning it has negative scaling factor) this still holds true: we just measure the angle from the center of the circles and in the direction "away" from the center of homothety, but with the same orientation (say, counterclockwise).
Inverse homothety.
Notice how this applies to the construction you described involving common tangents, and gives a generalization. The tangency points each have the same angle along their circle, and so does any other pair of homothetically corresponding points.
So, this can be used to give a construction which works in any possible case. Let $C_1$ and $C_2$ be two circles in any possible configuration, and $c_1$, $c_2$ their respective centers joined by the line $L$. To find the centers of homothety, construct the lines $L_1$ and $L_2$ which are perpendicular to $L$ through $c_1$ and $c_2$ respectively. Then mark the two intersections of $C_1$ and $L_1$ as $a_1$, $b_1$, and similarly with $a_2$, $b_2$.
Construct the line through $a_1$ and $b_1$ and find its intersection with $L$. This is one center of homothety. Joining $a_1$ and $b_2$ instead, that intersection with $L$ gives another center of homothety.
Construction with straightedge and compass.