Distance in vector space Suppose $k≧3$, $x,y \in \mathbb{R}^k$, $|x-y|=d>0$, and $r>0$. Then prove
(i)If $2r > d$, there are infinitely many $z\in \mathbb{R}^k$ such that $|z-x| = |z-y| = r$
(ii)If $2r=d$, there is exactly one such $z$.
(iii)If $2r < d$, there is no such $z$
I have proved the existence of such $z$ for (i) and (ii). The problem is i don't know how to show that there are infinitely many and is exactly one such z resectively. Plus i can't derive a contradiction to show that there is no such z for (iii). Please give me some suggestions

If there's a $z$ satisfying $|z-x|=r=|z-y|$ then by the triangle inequality, $d=|x-y|\le|z-x|+|z-y|=2r$
So if $d>2r$ there would've been no such $z$!
