This is a problem from baby rudin in Chapter 1, I've done all of them except this(problem 16).
suppose k>=3,x,y belongs to [R][/k],|x-y|=d>0,and r>0.prove:
(1)if 2r>d, there are infinitely many z belong to R(k) such that |z-x|=|z-y|=r.
(2)if 2r=d, there is exactly one such z.
(3)if 2r<d, there is no such z.
The Attempt at a Solution
I have done a part of this question. I can't prove (1). About (2), when 2r=d, let z=(x+y)/2, then |z-x|=|z-y|=d/2. However, I can't prove it is unique. About (3), suppose there exists such z, then by thereom 1.37 in baby rudin we have 2r=|z-x|+|z-y|>=|x-y|=d, which contradicts the fact that 2r<d.
May somebody help me!