I have a question about solving a system of geometric equations. I really hope someone here can help me, it's been several months since I try to solve the problem but without success. As I am not an expert in maths, I count on your help. Thank you in advance.
Problem: I have three circles $C_1, C_2, C_3$. For example (refer to this figure)
$$\begin{eqnarray*} C_1 :& (x+1)^2+(y−4)^2=9 \\ C_2 :& (x+4)^2+y^2=25 \\ C_3 :& (x−2)^2+y^2=16 \end{eqnarray*}$$
$N_2$ and $N_3$ are two points: $N_2$ is located at the intersection of $C_1$ and $C_2$, and is outside $C_3$. $N_3$ is located at the intersection of $C_1$ and $C_3$, and is outside $C_2$.
$T_1$, $T_2$ and $T_3$ are three points (defining the blue triangle in the figure): $T_1$ is located at $C_1$ (and inside $C_2$ and $C_3$). $T_2$ is located at the intersection of $C_2$ and the line that passes through $T_1$ and $N_2$. $T_3$ is located at the intersection of $C_3$ and the line that passes through $T_1$ and $N_3$.
Question: Given any three circles $C_1$, $C_2$, and $C_3$, is there a point $T_1$ (defined as above) such that $\text{distance}\space (T_1, T_2)=\alpha\cdot (\text{distance}\space (T_1, T_3))$ ($\alpha$ is some constant, for example equal to $1$)?
I tried Maple, it gives me the solution, but not the way how to calculate it. Any idea?
Edit: What I need is a proof of calculability. For example, in a system of linear equation with $x$ variables, we know that $x-1$ equations are needed, otherwise there is an infinity of possible solutions. For my system, I know how to prove that if a solution exists, it is necessarily unique. But I don't know an algorithm that can calculate it because the equations are not linear.
Thank you.