It is well known how to obtain the circumsphere of a simplex, given the coordinates of the vertices. But there is a rather nice expression that uses only the squared distances. One can read of the radius and barycentric coordinates directly from the Cayley-Menger matrix.
In 3D, if the inverse Cayley-Menger matrix is:
( -2r^2 a b c d ) ( a 0 C_12 C_13 C_14) ( b C_21 0 C_23 C_24) ( c C_31 C_32 0 C_34) ( d C_41 C_42 C_43 0 )
Then the barycentric coordinates of the circumsphere are (a,b,c,d) and r is the radius