- #1

jgens

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## Homework Statement

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

## Homework Equations

N/A

## The Attempt at a Solution

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

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!