315. Chords in a Sphere. Let three mutually perpendicular chords $A P B, C P D$, and $E P F$ be given in a certain sphere, passing through one point. Determine the radius of the sphere, given that $A P=2 a, B P=2 b, C P=2 c, D P=2 d, E P=2 e$, and $F P=2 f$.
315. If we choose a rectangular coordinate system, the axes of which are along the given chords, then the center of the sphere will be located at the point with coordinates ( $b-a, d-c, f-e$ ), where we can assume without loss of generality that $b \geqslant a, d \geqslant c$ and $f \geqslant e$. Then we will obtain[^37] the following relation for the radius of the given sphere:

$$
\begin{aligned}
R^{2} & =(b-a-2 b)^{2}+(d-c-0)^{2}+(f-e-0)^{2}= \\
& =a^{2}+b^{2}+c^{2}+d^{2}+e^{2}+f^{2}-2 e f
\end{aligned}
$$

(since $a b=c d=e f$ ).

It is worth noting that the result easily generalizes to the case of an $n$-dimensional sphere. In the special case of a circle ( $n=2$ ), only the squares remain, and the product of different segments disappears.

$[$ [M. M. Klamkin, M., 44, 55 (January 1971).]