The center of a circle in 3D can be calculated from equations of (squared) distances and coplanar requirement. This method does not suffer from producing too complicated formulas that are hard to simplify.
I am not sure why method in another OP question produces such convoluted formulas (although they are correct too).
Q = (2 - Sqrt[2])/4;
aa = {r/Sqrt[
3]*(Sqrt[2]*Q*(2 - U^2) + 2 - Sqrt[2]/4*c*U^2)/(Q*(2 - U^2) - 2),
r/Sqrt[1]*((c*(Q - 1) - 1)*U)/(Q*(2 - U^2) - 2),
r/Sqrt[6]*(Sqrt[2]*Q*(2 - U^2) + 2 -
Sqrt[2]/4*c*U^2)/(Q*(2 - U^2) - 2)};
bb = {r/Sqrt[
3]*(Sqrt[2]*Q*(1 - U^2) + 3 - 1/2*c*(1 + U^2))/(Q*(1 - U^2) - 2),
r/Sqrt[1]*(-(1 + c)*U)/(Q*(1 - U^2) - 2),
r/Sqrt[6]*(Sqrt[2]*Q*(1 - U^2)*(1 + 2*c*Q) + 2*c)/(Q*(1 - U^2) - 2)};
cc = {r/Sqrt[
3]*(Sqrt[2]*Q*(1 - U^2) + 1 +
c*(Sqrt[2]*Q*(1 - U^2) + 1))/(Q*(1 - U^2) - 2),
r/Sqrt[1]*(-(1 + c)*U)/(Q*(1 - U^2) - 2),
r/Sqrt[6]*(Sqrt[2]*Q*(1 - U^2) + 4 +
c*((1/2 + Q)*(1 - U^2) - 2))/(Q*(1 - U^2) - 2)};
circleCenter[a_, b_, c_] :=
Block[{dist =
SquaredEuclideanDistance @@@
Tuples[{{aa, bb, cc}, {{x, y, z}}}] /.
Abs -> Identity, eq},
eq = Equal @@ dist && CoplanarPoints[{a, b, c, {x, y, z}}];
SolveValues[eq, {x, y, z}]]][[1]]]
Factor[#, Extension -> Sqrt[2]] & /@
circleCenter[aa, bb, cc] // FullSimplify
