I don't know how to explain the arrangement, so I've attached an image. Forgive my notation, I'm not used to writing math problems.

The points that are black are known. The points that are red are unknown. $\displaystyle P_0T_{11}$ and $\displaystyle P_0T_{12}$ are tangent to BOTH circles. $\displaystyle P_1P_2$ is tangent only to circle 2 (the larger circle).

Conceptually, I think this is easy. The center of circle 2 ($\displaystyle C_2$) lies on $\displaystyle P_0P_2$, and the radius is such that it is tangent to both $\displaystyle C_1$ and $\displaystyle C_2$. Without any other conditions, there would be an infinite number of solutions. However, there is an additional line that bounds circle 2. So, unless I am mistaken, there is now only one solution.

However, I don't know what I'm doing wrong. I've tried using similar triangles, but because neither $\displaystyle C_2$ nor the radius for circle 2 is known, I haven't been able to determine the ratio of the two triangles. In fact, everything I've tried seems to require a system of equations.

So I tried the following, where $\displaystyle d(\textbf{x}_0,\textbf{x}_1)$ is the distance formula between points $\displaystyle \textbf{x}_0$ and $\displaystyle \textbf{x}_1$:

$\displaystyle d(P_0,T_2_1)=d(P_0,T_2_2)$

$\displaystyle d(P_1,T_2_2)=d(P_1,T_3)$

$\displaystyle d(P_0,T_2_2)+d(T_2_2,P_1)=d(P_0,P_1)$

However, when I write the distance formulas for three dimensions, I end up with nine unknowns; three for each of the following points, $\displaystyle T_2_1$, $\displaystyle T_2_2$, and $\displaystyle T_3$.

So, I tried an additional three equations based on the distance of the center of circle 2 to each of the bounding lines using the following formula:

$\displaystyle d=\dfrac{\mid(\textbf{x}_0-\textbf{x}_1)\times(\textbf{x}_0-\textbf{x}_2)\mid}{\mid\textbf{x}_2-\textbf{x}_1\mid}$

I needed three more equations, so I tried parameterizing the equation for $\displaystyle P_0P_1$, yielding:

$\displaystyle x_{T_{22}}=x_{P_0}+t(x_{P_1}-x_{P_0})$

$\displaystyle y_{T_{22}}=y_{P_0}+t(y_{P_1}-y_{P_0})$

$\displaystyle z_{T_{22}}=z_{P_0}+t(z_{P_1}-z_{P_0})$

where $\displaystyle t=\dfrac{d(P_0,T_{22})}{d(P_0,P_1)}$

However, this led to a binomial equation in x, y and z. So, instead I tried using the equation for each of the three bounding lines, since the respective tangent point is a solution to the corresponding line equation. Unfortunately, I'm not sure if I'm not solving the system of equations correctly, or if my equations are not independent from each other.

Again, I feel like there should be an elegant solution to this. Perhaps something similar to finding the tangent points of a circle inscribed inside a triangle.

Any help would be greatly appreciated.