Let $A,B,C$ be three distinct points in space, $(A)$ the sphere with center $A$ and radius $r$. Let $E$ be the set of numbers $R>0$ for which there is a sphere $(H)$ with center $H$ and radius $R$ such that $B$ and $C$ are outside the sphere, and the points of the sphere $(A)$ are strictly inside it. (a) Suppose that $B$ and $C$ are on a line with $A$ and strictly outside $(A)$. Show that $E$ is nonempty and bounded, and determine its supremum in terms of the given data. (b) Find a necessary and sufficient condition for $E$ to be nonempty and bounded (c) Given $r$, compute the smallest possible supremum of $E$, if it exists.