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Math Help - Euclidean Space

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by EyeHeartAdrianT View Post
    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.
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