Circumcenter of Tetrahedron (in 4D) I am trying to calculate the circumcenter of a tetrahedron in 4 dimensional space. I was hoping for some concrete mathematical formula which can make this calculation more accurate.
 A: Let the points be $(a1,a2,a3,a4),(b1,b2,b3,b4),(c1,c2,c3,c4),(d1,d2,d3,d4)$.
The circumcentre is in the hyperplane, so it is a convex combination of the points 
$$CC = wA+xB+yC+(1-w-x-y)D$$
The distances to the points are the same.
$$|(1-w)AD-xBD-yCD|^2\\=|-wAD+(1-x)BD-yCD|^2\\=|-wAD-xBD+(1-y)CD|^2\\=|-wAD-xBD-yCD|^2$$
Subtract the last expression from the first three, to get three linear equations in $w,x,y$:
$$\left[\begin{array}{ccc}2AD.AD&2AD.BD&2AD.CD\\2BD.AD&2BD.BD&2BD.CD\\2CD.AD&2Cd.BD&2CD.CD\end{array}\right]\left[\begin{array}{c}w\\x\\y\end{array}\right]=\left[\begin{array}{c}AD.AD\\BD.BD\\CD.CD\end{array}\right]$$
For example, $2AD.BD=2(d-a).(d-b)$.  Invert the 3x3 matrix, to find $w,x$ and $y$.
A: This goes well beyond your question, 
but this paper was posted to the arXiv just last week:

Jean-Daniel Boissonnat, Ramsay Dyer, Arijit Ghosh.
  "A probabilistic approach to reducing the algebraic
  complexity of computing Delaunay triangulations."
  (arXiv abstract).
  
  Computing Delaunay triangulations in $\mathbb{R}^d$ involves evaluating the so-called in_sphere predicate that determines if a point $x$ lies inside, on or outside the sphere circumscribing $d+1$ points $p_0,\ldots,p_d$. This predicate reduces to evaluating the sign of a multivariate polynomial of degree $d+2$ in the coordinates of the points $x,p_0,\ldots,p_d$. [...]




Searching on the phrases in_sphere or in-sphere or insphere will access
the substantial literature on the topic.
A: By virtue of https://math.stackexchange.com/a/2174952/36678, the quadriplanar coordinates of the circumcenter are expressed only in dot-products, and are thus trivially extended to any dimension.
Given the quadriplanar coordinates $q_i$, the circumcenter is trivially retrieved by
$$
C = \frac{\sum_i q_i \alpha_i X_i}{\sum_i q_i \alpha_i}
$$
where $\alpha_i$ are the respective face areas. (That is to say: You'll only need the dot-product expressions from https://math.stackexchange.com/a/2174952/36678).
