For any two vertices, for example, 1 and 2, the center is equidistant from them. Therefore it lies in the (n-1)-dimensional subspace passing through the midpoint of the segment x1—x2 (x1 — vector of the first vertex) and perpendicular to it. The equation of this plane:
Because x1-x2 is the direction vector of the edge (perpendicular to the plane). The subscripts denote the number of coordinates, and the upper vertices of the simplex.
The center uniquely determines the intersection of the n planes. For example, we choose plane 1—i, where i varies from 2 to (n+1). Then it will be
The obtained inhomogeneous linear system is relatively
which is allowed (by any method).
P. S. the Correctness is not guaranteed. But look forward to it.