I have the following quite simply looking equation
$$ \frac{c_1 d_1}{c_2+c_3}=\frac{c_2 d_2}{c_1+c_3} = \frac{c_3 d_3}{c_2+c_1}$$
and I want to solve it for $c_1, c_2,c_3$. I know that these equations are homogen in $c$, hence I'm only looking for a solution up to scaling.
All $c_i$ are positive and I know that there exists a (unique up to scaling) solution. The $d_i$ come from a more complicated equation and might in some special cases cause trouble. I'm only interested in the general solution, not in special cases, that is the cases where $d_i$ cause difficulties can be ignored.
Because the equation looks simple I expect a "simple" symmetric solution (I know that this doesn't have to be the case).
So far I tried solving the first equation for $c_3$ which is easy as it is linear. Then I inserted this into the other remaining equation, this however results in a polinomial of degree 3. With somewhat ugly coefficients. I tried to solve this polinomial but it gets quite dirty, so I asked a mathematical algebra tool (python sympy), it's no been computing since 10 minutes without a result, so I doubt it will come up with a good solution.
Is there an easy way to guess the solution?
I heard of the Groebner basis, but don't know how to apply it (that is I don't even know exactly what it is, just that it's a tool to solve polynomial equations in multiple dimensions). Can one apply this theory here?
Let \begin{align} \frac{c_1d_1}{c_2+c_3}&=k \tag{1}\label{1} ,\\ \frac{c_2d_2}{c_3+c_1}&=k \tag{2}\label{2} ,\\ \frac{c_3d_3}{c_1+c_2}&=k \tag{3}\label{3} . \end{align}
Solution of the system \eqref{1},\eqref{2} gives
\begin{align} c_1 &= \frac{kc_3(k+d_2)}{d_1d_2-k^2} \tag{4}\label{4} ,\\ c_2 &= \frac{kc_3(k+d_1)}{d_1d_2-k^2} \tag{5}\label{5} , \end{align}
substitution of \eqref{4},\eqref{5} into \eqref{3} gives cubic equation
\begin{align} 2k^3+(d_1+d_2+d_3)k^2-d_1d_2d_3&=0 \end{align}
which has at least one real root. Given that, \eqref{4}, \eqref{5} express $c_1,c_2$ in terms of $d_1,d_2,d_3$ and arbitrary $c_3$.