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Every circle has form |z-a|=k|z-b|

  1. Nov 11, 2011 #1
    We can express any circle in the complex plane as |z-a|=k|z-b| where a and b are distinct complex numbers, k > 0 and [itex]k \not= 1.[/itex]

    Is there an elegant way of showing this fundamental property of the complex plane to be true?
     
  2. jcsd
  3. Nov 11, 2011 #2

    mathman

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    It doesn't look elegant, but squaring both sides leads to a straightforward proof.
     
  4. Nov 12, 2011 #3

    AlephZero

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    Google for "circle of Apollonius". This geometrical result was known centuries before complex numbers and analytic geometry were invented.
     
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