I reduced a system of non-linear equations to a single equation for $c$:
$$\left(\frac{x-c}{y-c}\right)^{w - u} = \left(\frac{x-c}{z-c}\right)^{v-u}$$
I need to solve this equation for $c \notin \{ x, y, z \}$ knowing $x, y, z, u, v, w \in \mathbb{R}$ where $x \neq y, x \neq z, y \neq z$ and $u \neq v, u \neq w, v \neq w$.
Is there any way to solve this equation directly? I would like to avoid iterative algorithms as part of this problem. If there is no direct way to solve the equation, which algorithm would you suggest?
If we multiply all the denominators away, we get
$$(x-c)^{w-u}(z-c)^{v-u}=(x-c)^{v-u}(y-c)^{w-u}$$
For simplicity, let $w-u=a$ and $v-u=b$.
$$(x-c)^a(z-c)^b=(x-c)^b(y-c)^a$$
Before I tackle $a,b\in\mathbb R$, let's look at $a,b\in\mathbb N$.
Unfortunately, for $a,b\ge5$, there exists no closed form solution of $c$ in terms of radicals for general $x,y,z\in\mathbb R$ due to the Abel-Ruffini theorem. It follows that we can't expect a closed form for $a,b\in\mathbb R$ either...
To be suggestive, Newton's method should work nicely. Here, we have
$$c_{n+1}=c_n-\frac{(x-c_n)^{a-b}(z-c_n)^b-(y-c_n)^a}{\left(\frac{a-b}{x-c_n}+\frac b{x-c_n}\right)(x-c_n)^{a-b}(z-c_n)^b-a(y-c_n)^{a-1}}$$