1. ## Euclidean Space

Hello, I have been having trouble with the following problem:

**E means epsilon**

Suppose k >= 3, x,y E R^k, |x - y| = d>0, and r>0. Prove that if 2r > d, there are infinitely many z E R^k such that |z - x| = |z - y| = r.

I'm not sure where to go with this problem...I've tried using the inequalities defined for Euc. Spaces in my text i.e.:

|z - y| <= |z - x| + |x - y|
|x - y| <= |x - z| + |z - y|
etc.

But I'm not sure if manipulating these inequalities is going to get me where I need to go. Any help with this problem would be much appreciated thank you.

Hello, I have been having trouble with the following problem:

**E means epsilon**

Suppose k >= 3, x,y E R^k, |x - y| = d>0, and r>0. Prove that if 2r > d, there are infinitely many z E R^k such that |z - x| = |z - y| = r.

I'm not sure where to go with this problem...I've tried using the inequalities defined for Euc. Spaces in my text i.e.:

|z - y| <= |z - x| + |x - y|
|x - y| <= |x - z| + |z - y|
etc.

But I'm not sure if manipulating these inequalities is going to get me where I need to go. Any help with this problem would be much appreciated thank you.
Hint. Let w be the midpoint between x and y. Then x, w, z will have to be the vertices of a right-angled triangle with hypotenuse r.