9. 2a.(USA 3) Determine the radius of a sphere $S$ that passes through the centroids of each face of a given tetrahedron $T$ inscribed in a unit sphere with center $O$. Also, determine the distance from $O$ to the center of $S$ as a function of the edges of $T$.
9. Let $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ denote the vectors $\overrightarrow{O A}, \overrightarrow{O B}, \overrightarrow{O C}, \overrightarrow{O D}$ respectively. Then $|\vec{a}|=|\vec{b}|=|\vec{c}|=|\vec{d}|=1$. The centroids of the faces are $(\vec{b}+\vec{c}+\vec{d}) / 3$, $(\vec{a}+\vec{c}+\vec{d}) / 3$, etc., and each of these is at distance $1 / 3$ from $P=$ $(\vec{a}+\vec{b}+\vec{c}+\vec{d}) / 3$; hence the required radius is $1 / 3$. To compute $\mid P$ as a function of the edges of $A B C D$, observe that $A B^{2}=(\vec{b}-\vec{a})^{2}=$ $2-2 \vec{a} \cdot \vec{b}$ etc. Now
$$
\begin{aligned}
P^{2} & =\frac{|\vec{a}+\vec{b}+\vec{c}+\vec{d}|^{2}}{9} \\
& =\frac{16-2\left(A B^{2}+B C^{2}+A C^{2}+A D^{2}+B D^{2}+C D^{2}\right)}{9} .
\end{aligned}
$$