What is the internal tangent circle problem for three given circles?

[DaalChawal]

Help in circle touching internally all these three circles.

 

[Opalg]

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In the diagram, the three given circles are the red one with centre $(6,8)$ and radius $5$, the blue one with centre $(-12,9)$ and radius $10$, and the green one with centre $(0,-8)$ and radius $3$. It should be easy to see that the circle which touches all three of them externally is the purple one with centre at the origin and radius $5$. But it is altogether harder to locate the black circle which touches all three of them internally. In fact, I doubt whether it is possible to find an exact solution. The best I can do is a numerical approximation that gives its centre as $(-5.912,5.002)$ and its radius as $17.283$.