In the article "Common Tangent Circles of Three Tangent Circles", I discussed using inversion to construct common tangent circles for three tangent circles. At that time, it was required that the three circles be pairwise tangent. Upon further reflection, I have found that this condition is not strictly necessary; it is sufficient for one of the three circles to be tangent to the other two.
Furthermore, the conditions can be even more relaxed: as long as two of the three circles are tangent, a common tangent circle satisfying the conditions can exist even if the third circle is not necessarily tangent to the others. As for the necessary and sufficient condition for the third circle, it can be seen from its inverse image: the inverse image of the third circle must intersect the region between the two parallel lines.
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