Similitude Center

Also called a Self-Homologous Point. If two Similar figures lie in the plane but do not have parallel sides (they are not Homothetic), there exists a center of similitude which occupies the same homologous position with respect to the two figures. The Locus of similitude centers of two nonconcentric circles is another circle having the line joining the two homothetic centers as its Diameter.

There are a number of interesting theorems regarding three Circles (Johnson 1929, pp. 151-152).

1. The external similitude centers of three circles are Collinear.

2. Any two internal similitude centers are Collinear with the third external one.

3. If the center of each circle is connected with the internal similitude center of the other three [sic], the connectors are Concurrent.

4. If one center is connected with the internal similitude center of the other two, the others with the corresponding external centers, the connectors are Concurrent.

References

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 19-27 and 151-153, 1929.