Given an arbitrary number of points, how do you find an equidistant center? Given an arbitrary set of points on a Cartesian coordinate plane, is there a generalized formula to find the closest point that is equidistant from all the given points? 
My first guess was finding the centroid, which is fairly easy to calculate, but the centroid of a polygon isn't equidistant from all its vertices.

There is no point that is equidistant from 4 or more points in general position in the plane, or $n+2$ points in $n$ dimensions.
Criteria for representing a collection of points by one point are considered in statistics, machine learning, and computer science.  The centroid is the optimal choice in the least-squares sense, but there are many other possibilities.
Added to answer the comment: 
The centroid is the point $C$ in the the plane for which the sum of squared distances $\sum |CP_i|^2$ is minimum.  One could also optimize a different measure of centrality, or insist that the representative be one of the points (such as a graph-theoretic center of a weighted spanning tree), or assign weights to the points in some fashion and take the centroid of those.
