I'm working on a problem that states if $k\geq 3$, $x,y\in\mathbb{R}^k$, $|x-y|=d>0$, and $r>0$, then
(a) If $2r>d$, there are infinitely many $z\in\mathbb{R}^k$ such that $|z-x|=|z-y|=r$.
(b) If $2r=d$, there is exactly one such $z$.
Intuitively and geometrically, this is obvious. The $z$ are all points distance $r$ from both $x$ and $y$, so they are the surfaces of the balls of radius $r$ centered around $x$ and $y$. If $2r>d$, then the radii are greater than half the distance between $x$ and $y$, and thus the balls will intersect in some surface, and if $k\geq 3$, then there will be infintely many points on this surface.
If $2r=d$, these balls will intersect in exactly one point of tangency.
Is there a way to more rigorously show these results without appealing to geometric intuition? For example, is there some algebraic system that you could exhibit in each case and say, here there are infinitely many solutions or there is exactly one solution, respectively? Thanks.