MathDB

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.

Problem Statement