$p$,

1.  $C_1$ located at $\alpha(p)$ and $C_2$ at $\beta(p)$, and

2.  $C_1$ located at $\beta(p)$ and $C_2$ at $\mu(p)$.

The case in which $C_1$ is at $\alpha(p)$ and $C_2$ at $\mu(p)$ is equivalent to the first configuration.

The expressions for the distances $\delta_{\alpha}(p)$, $\delta_{\beta}(p)$ and $\delta_{\mu}(p)$, and angles $\cos(\alpha,\beta)$ and $\cos(\beta,\mu)$ are $$\begin{aligned}
\delta_{\alpha}(p) & = & \sqrt{|a|^2 + 4(y^2+z^2)} \nonumber \\
\label{DistFunc}
 \delta_{\beta}(p) & = & \sqrt{|b|^2 + 4x^2 + (2z-|c|)^2} \\
   \delta_{\mu}(p) & = & \sqrt{|b|^2 + 4x^2 + (2z+|c|)^2} 
\nonumber\end{aligned}$$ and $$\begin{aligned}
\cos(\alpha,\beta) & = & \frac{4z^2 - 2(|a|x + |b|y + 
|c|z)}{\delta_{\alpha}(p) \delta_{\beta}(p)} \nonumber \\
\label{CosAngle} & & \\
   \cos(\beta,\mu) & = & \frac{4x^2 + 4z^2 +|b|^2 - 
|c|^2}{\delta_{\beta}(p) \delta_{\mu}(p)} \nonumber\end{aligned}$$

For the first configuration one has $\delta_{\alpha}(p) = 
d_1$, $\delta_{\beta}(p) = d_2$ and $\cos(\alpha,\beta) = 
\cos\theta$, thus yielding the equations $$\begin{aligned}
                  y^2 + z^2 & = & \frac{1}{4} (d_1^2 - |a|^2) 
\nonumber \\ 
\label{EqFirstG6}
        4x^2 + (2z - |c|)^2 & = & d_2^2 - |b|^2 \\
4z^2 -2(|a|x + |b|y + |c|z) & = & d_1d_2\cos\theta \; . \nonumber\end{aligned}$$ This is a system of three quadratic equations with six unknowns, the three coordinates $(x,y,z)$ of the point $p$, and the three coordinates $(|a|,|b|,|c|)$ in 