Solutions to Apollonius' problem are related by circle inversion.  Here, the three given circles are shown in black, with their radical center shown as a large orange point, whereas the two solutions are shown as pink circles.   The orange orthogonal circle centered on the radical center intersects the three given circles perpendicularly (orthogonally), so that the given circles map to themselves under inversion  in the orange circle.  The two pink circles map to each other, as do their corresponding tangent points (blue) with the given circles.  The green lines also map to themselves, since they pass through the center of inversion.
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WillowW