Given that the question asks for a method to find a point equidistant to a given set of points and the accepted answer claims that the centroid provides the answer in the "least-squares sense", it seems appropriate to present the actual least-squares solution.
Let the cost-function be:
$$
\tilde{y}_i = \lvert\lvert x_i-\hat{x}_c\rvert\rvert_2 - \hat{r}_c
$$
Where $x_i$ is the Cartesian coordinate of the $i^{th}$ point in the set, $\hat{x}_c$ is the estimated "equidistant center", and $\hat{r}_c$ is the estimated distance between $\hat{x}_c$ and all other points $x_i$.
The Jacobian of $\tilde{y}_i$ with respect to the desired state $\begin{bmatrix} \hat{x}_c \\ \hat{r}_c \end{bmatrix}$ is:
$$
H_{i,12} = -\frac{(x_i - \hat{x}_c)}{\lvert\lvert x_i - \hat{x}_c\rvert\rvert_2}
$$
$$
H_{i,3} = -1
$$
Where $H$ is a $N_p \times 3$ matrix ($N_p$ is the number of 2D Cartesian points), $H_{i,12}$ is the first two columns of the $i^{th}$ row, and $H_{i,3}$ is the third column of the $i^{th}$ row.
Let:
$$
\hat{\mathbf{x}} = \begin{bmatrix}
\hat{x}_c \\
\hat{r}_c \\
\end{bmatrix}
$$
be the stacked state vector. A single least-squares update looks like:
$$
\hat{\mathbf{x}}_{k+1} = \hat{\mathbf{x}}_k - (H^T H)^{-1} H^T \tilde{y}
$$
The least-squares estimate to the question can be calculated by initializing $\mathbf{x}_{k=0}$ with an initial guess (perhaps involving the centroid) and iteratively updating $\tilde{y}$ and $\hat{\mathbf{x}}$.
This is the answer in the least-squares sense.
This set of 6 points:
$x = \lbrace[443.292, 397.164], [355.53, 349.168], [326.975, 253.249], [562.139, 186.385], [375.017, 165.456], [473.424, 137.604]\rbrace$
has a ``least-squares equidistant center'' of $[457.9097, 267.1183]$ with a point separation of $131.1977$.
Note that the given points are 6 of 8 equally spaced points around a circle (i.e., 2 are missing) and the least-squares solution recovers both the center of the circle and its radius.