The Common Tangent Circle of Three Tangent Circles: A Supplement
In the article The Common Tangent Circle of Three Tangent Circles, I discussed constructing the common tangent circle of three tangent circles by means of inversion; at that time I required that the three circles be pairwise tangent to one another. Having thought about it further since, I realized this condition isn't actually necessary — it suffices that just one of the three circles be tangent to each of the other two.more
In fact, one can relax the condition even further: among the three circles, only two need be tangent to each other, while the third circle need not be tangent to either of them, and a common tangent circle satisfying the required condition may still exist. As for the necessary and sufficient condition on this third circle, it can be seen from its inverted image: the inverted image of the third circle must intersect the region lying between the two parallel lines.
Common tangent circle of three circles 1
Common tangent circle of three circles 2
Translated automatically with claude-sonnet-5; all equations are reproduced verbatim from the source. Copyright remains with the original author.