# Euclidean spaces

## Homework Statement

This is a problem from baby rudin in Chapter 1, I've done all of them except this(problem 16).
suppose k>=3,x,y belongs to [R][/k],|x-y|=d>0,and r>0.prove:
(1)if 2r>d, there are infinitely many z belong to R(k) such that |z-x|=|z-y|=r.
(2)if 2r=d, there is exactly one such z.
(3)if 2r<d, there is no such z.

## The Attempt at a Solution

I have done a part of this question. I can't prove (1). About (2), when 2r=d, let z=(x+y)/2, then |z-x|=|z-y|=d/2. However, I can't prove it is unique. About (3), suppose there exists such z, then by thereom 1.37 in baby rudin we have 2r=|z-x|+|z-y|>=|x-y|=d, which contradicts the fact that 2r<d.

May somebody help me!

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OMG,will somebody tell me why there is not anybody replied?

Mark44
Mentor
OMG,will somebody tell me why there is not anybody replied?
If someone tells you why no one has replied, then someone would have replied, making the need for an answer to your question unnecessary. For the three parts of your question, you are dealing with three points in Rk: x, y, and z.

Draw a picture of x and y at a distance of d apart. Now add a point z. For part 2, it's pretty clear from the geometry that there is only one place that z could be. For part 3, it's also pretty clear that there could not be any point z that satisfies the given condition.

For part 1, the geometry shows that there are an infinite number of possible locations for z.

For the three parts of your question, you are dealing with three points in Rk: x, y, and z.

Draw a picture of x and y at a distance of d apart. Now add a point z. For part 2, it's pretty clear from the geometry that there is only one place that z could be. For part 3, it's also pretty clear that there could not be any point z that satisfies the given condition.

For part 1, the geometry shows that there are an infinite number of possible locations for z.[/QUOTE]
I appreciate it very much. However, I don't think it is an analytic way. Any way, thank you for so kind to a noob.

Mark44
Mentor
But drawing a sketch shows you the geometry, which should get you thinking about how you would write up a more analytical explanation.