Question 13
(USA 6; IMO 2) Given any point $P$ in the interior of a sphere with radius $R$, three mutually perpendicular segments $PA,PB,PC$ are drawn terminating on the sphere and having one common vertex in $P$. Consider the rectangular parallelepiped of which $PA,PB,PC$ are coterminal edges. Find the locus of the point $Q$ that is diagonally opposite $P$ in the parallelepiped when $P$ and the sphere are fixed.