3D coordinates of circle center given three point on the circle. Given the three coordinates $(x_1, y_1, z_1)$, $(x_2, y_2, z_2)$, $(x_3, y_3, z_3)$ defining a circle in 3D space, how to find the coordinates of the center of the circle $(x_0, y_0, z_0)$?

In  this formula
the center $O$ of
a circumscribed circle of $\triangle ABC$
is expressed as a convex combination of its vertices
in terms of
coordinates $A,B,C$ and corresponding side lengths $a,b,c$,
suitable for both 2d and 3d:
\begin{align}
O&=
A\cdot \frac{a^2\,(b^2+c^2-a^2)}{((b+c)^2-a^2)(a^2-(b-c)^2)}
\\
&+B\cdot \frac{b^2\,(a^2+c^2-b^2)}{((a+c)^2-b^2)(b^2-(a-c)^2)}
\\
&+C\cdot \frac{c^2\,(b^2+a^2-c^2)}{((b+a)^2-c^2)(c^2-(b-a)^2)}
.
\end{align}
