How to construct center of homothety for two circles which overlap

Question

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.

Answer 1

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.

Answer 2

In the case of two circles that are not disjoint, I think that you only have one homothety center, which is the one found by the intersecction of the external tangents. The reason for that is that when you have two circles that are not disjoint the internal tangent no longer exists. In case you don't know how to find the external tangent of two cercles (it doesn't matter if they are disjoint or not) with compass and straightedge, here you have images and an explanation that shows you how to do it.

http://jwilson.coe.uga.edu/emt669/Student.Folders/Kertscher.Jeff/Essay.3/Tangents.html

Basically, you draw a line between the two centers, you find the medium point of this line and draw a circle centered there that touches both cirle centers using the compass. After that, you draw a circle centered in the center of the biggest circle with a radious equal to R= R(big circle)-R(small circle). You draw a line from the center of the biggest circle to the point that you find on the intersection of the two new circles that you've drawn, and follow this line until it intersects with the biggest circle. This point is the first tangent point. If you now draw a parallel line to this one that passes through the center of the small circle, you'll have the second tangent point in the intersection of this line with the small circle. If you draw a line joining both tangent points and make it intersect with the line that joins both circle centers, there you'll have your homothetic center.

For more information, you may want to take a look at this:

https://commons.wikimedia.org/wiki/Category:Homothety_in_circles

Maybe you are looking not for homothety centers but antihomothety centers. I hope I've been able to help you, and don't hesitate to tell me if not, i'll try to do my best!