Given two points inside the unit circle, $(x_1, y_1)$ and $(x_2, y_2)$, let $C_1$ and $C_2$ be the circles with centers at those points, respectively, which are internally tangent to the unit circle. If $C_1$ and $C_2$ are externally tangent to each other, how can I find the center $(x, y)$ of the circle $C$ which is internally tangent to the unit circle and externally tangent to both $C_1$ and $C_2$?
I know that the radii of the circles will satisfy the Descartes Circles Theorem, and that the unit circle must be the external Soddy Circle for $C$, $C_1$, and $C_2$, so I could theoretically back-solve for $(x,y)$ knowing that $(0,0)$ is the isoperimetric point of the triangle with vertices at each circle's center, but that could get real ugly real quick.
If $C$, $C_1$, and $C_2$ have radii $r$, $r_1$, and $r_2$, respectively, then I think that $(x,y)$ is the intersection of the circle with radius $1-r$ and the hyperbola with focii at $(x_1, y_1)$ and $(x_2, y_2)$ and difference in distances $|r_1-r_2|$, but that doesn't help too much for determining the coordinates $(x, y)$.
Is there a better way of doing this?