I am trying to get a least-square circles fit. That requires $(x_c, y_c,r)$. I know that for that the system $S = \sum_{i=1}^n{[(x_i-x_c)^2 + (y_i - b)^2 - r^2]^2}$. The way to do this is to equate the partial derivatives of the unknowns $(\partial S / \partial x_c, \partial S / \partial y_c, \partial S / \partial r)$ to $0$ and solve the system of equations.
What to do when $r$ is known? My solution gives a system with two quadratic equations of order $(x_c^3, y_c^3)$ and I was wondering if there is an easier way. Should I move to a numerical solution?
As you wrote, you have $n$ data points $(x_,y_i)$ and you want to minimize
$$S = \sum_{i=1}^n \Big[(X-x_i)^2 + (Y-y_i)^2 - R^2\Big]^2$$ As usual, we need reasonable estimates of $(X,Y)$.
In a first step, consider the $n$ equations $$F_i=(X-x_i)^2 + (Y-y_i)^2 - R^2=0$$ and buid the $\frac{1}2 n(n-1)$ equations $$G_{ij}=F_i-F_j=2(x_j-x_i)X+2(y_j-y_i)Y=(x_j^2+y_j^2)-(x_i^2+y_i^2)$$ where $i$ varies from $1$ to $(n-1)$ and $j$ from $(i+1)$ to $n$. This is a very simple problem.
For illustration, using the data taken from this paper (page $2$). this gives $X_0=3.06030$ and $Y_0=0.743607$ which are almost exactly the results obtained using Gruntz's procedure.
Now, we can go back to the original problem which means that we need to solve $$\frac{\partial S}{\partial X}=4\sum_{i=1}^n (X-x_i)\Big[(X-x_i)^2 + (Y-y_i)^2 - R^2\Big]=0$$ $$\frac{\partial S}{\partial Y}=4\sum_{i=1}^n (Y-y_i)\Big[(X-x_i)^2 + (Y-y_i)^2 - R^2\Big]=0$$ which is very simple using Newton-Raphson method starting with $(X_0,Y_0)$ as initial guesses.
Using these values for a few values of $R$
$$\left( \begin{array}{ccc} R & X & Y \\ 3.7 & 3.11165 & 0.800459 \\ 3.8 & 3.10106 & 0.788820 \\ 3.9 & 3.08933 & 0.775878 \\ 4.0 & 3.07628 & 0.761420 \\ 4.1 & 3.06172 & 0.745189 \\ 4.2 & 3.04537 & 0.726870 \\ 4.3 & 3.02693 & 0.706070 \end{array} \right)$$