center of circumscribed n-sphere

Question

Mathematics

Given the coordinates of the vertices of the simplex in Euclidean n-dimensional space,
how to calculate the coordinates of the center of the n-dimensional sphere circumscribed around it?

Answer 1

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 middle of the segment x1-x2 (x1 is the vector of the first vertex) and perpendicular to it. The equation for such a plane is:

Because x1-x2 is the direction vector of the edge (the normal to the plane). The subscripts denote the number of the coordinate, and the superscripts denote the vertices of the simplex.
The center uniquely defines the intersection of n planes. For example, let's choose planes 1—i, where i varies from 2 to (n+1). Then

an inhomogeneous linear system with respect to

which is resolved (by any method) will be obtained.
PS Correctness is not guaranteed. But I hope for her.