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

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The internal tangent circle problem involves finding a circle that touches three given circles internally. The three circles are defined with specific centers and radii, with one circle touching them externally identified as having its center at the origin and a radius of 5. However, locating the internal tangent circle is more complex, and an exact solution may not exist. A numerical approximation suggests the internal circle's center is at approximately (-5.912, 5.002) with a radius of about 17.283. This highlights the challenges in solving the internal tangent circle problem for three given circles.
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Help in circle touching internally all these three circles.
 
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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 $17.283$.
 
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