For some tasks to prove I need to know if there always exists a circle which is tangent to three given circles but contains none of the three given circles. It is ensured to have middle radius smaller than left and right radius. I know there is a problem named Apollonius' Problem which is closely related to it. I searched a bit about this on internet. But couldn't find straight answer to my question. That's why I am asking here. It'd be great if any one could provide the link of proof in favor of or contrary to the above statement.
2026-04-23 05:44:56.1776923096
Circle touching three circles
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Not my best artwork, anyway, three original circles in deep purple with shading; two light blue mutual tangent circles, two light green, four orange. Should anyone have trouble seeing which colors go where, both green and blue are left-right symmetric, all three circles are inside the light blue circles/arcs, only the large central purple circle is inside the two green circles.