Suppose $k \geq 3$, $x$, $y \in \mathbb{R}^k$, $|x-y| = d > 0$, and $r > 0$. Then how to prove the following assertions?
(a) If $2r > d$, then there are infinitely many $z \in \mathbb{R}^k$ such that $$ | z - x | = | z -y | = r.$$
(b) If $2r = d$, then there is exactly one such $z \in \mathbb{R}^k$ for which $$ | z - x | = | z -y | = r.$$
(c) If $2r < d$, then there is no $z \in \mathbb{R}^k$ such that $$ | z - x | = | z -y | = r.$$
How must these statements be modified if $k$ is $2$ or $1$?
Try to think geometrically. What is the collection of all point $z$ such that $|z - x| = r$ for some fixed $r$?
To get some intuition (although you must be careful, particularly given the last question that you ask), it might help to draw this out for $k = 2$ and then try to understand how this is related in the case $k > 2$.