Steiner ellipse extended to 3D

Consider a regular tetrahedron with its bottom face on the $xy$ -plane, its top vertex at $C$ , and its center at $D$ . By the properties of a regular tetrahedron , the center will be $\cfrac{r}{R + r} = \cfrac{r}{3r + r} = \cfrac{1}{4}$ of the way up the height (where $R$ is the radius of the circumsphere and $r$ is the radius of the insphere, and $R = 3r$ ).

Now stretch the regular tetrahedron and its circumsphere so that $C$ moves to $C'(0, h)$ . The stretched circumsphere is now the ellipsoid of minimum volume with the given requirements (the Steiner ellipsoid?), and its new center will still be $\cfrac{1}{4}$ of the way up its height.

Therefore, when $h = 24$ , the $z$ -coordinate of the ellipsoid's center is $\cfrac{1}{4}h = \cfrac{1}{4}\cdot 24 = \boxed{6}$ .