> 
> Let $x,y \in \mathbb{R}^k$ ($k\geq 3$), $|x-y|=d>0$ and $r>0$.
>  Prove that if $2r>d$, then there are infinitely many $z\in \mathbb{R}^k$ such that $|z-x|=|z-y|=r$.
> 
> 
> 

Here's what I have proved; 

1. The existence of such $z$, and
2. $|z-x|=|z-y|=r$ iff $(z-(x+y)/2)\cdot(x-y)=0$ and $|z-(x+y)/2|=\sqrt{r^2 - d^2/4}$

I know exactly what's happening here and that there are infinitely many such $z$, but cannot show this logically. This prob is on 'analysis by rudin' so no topology please..

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Edited: Only thing i need to prove here is that 'There exist infinitely many $d\in \mathbb{R}^k$ such that $(x-y)•d=0$ and $|d|=1$.'