# Euclidean Spaces

Gold Member

## Homework Statement

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

N/A

## The Attempt at a Solution

Well, I figure that no matter how large we choose $n$, it should always be possible to reduce the problem to the case when $n=3$. Since we just have two points in $\mathbb{R}^n$, these can just be represented by a one dimentional line. Then, we can always cut the line at $(x-y)2^{-1}$ with a plane which is perpendicular to $x-y$. Clearly, any point on this plane is at an equal distance from $x$ and $y$; moreover, there must be some circle on this plane such that all the points on that circle are a distance of $r$ from $x$.

However, this reasoning is really informal (and not necessarily correct) so I was wondering if I could formalize this idea or if I'm approaching this problem with the completely wrong mindset (I think that I probably am). Any help is appreciated!