 

Finding the circumcenter of a tetrahedron is discussed on page 33 in

CONTOURING: A guide to the analysis and display of spatial data,
by Dave Watson, Pergamon Press,  1992, ISBN 0 08 040286 0, 321p.

Each pair of tetrahedral vertices define a plane which is a 
perpendicular bisector of the line between that pair.  Express each
plane in the form Ax + By + Cz = D
and solve the set of simultaneous equations from any three of those
planes that have a vertex in common (all vertices are used).  
The solution is the circumcenter.
