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Thread: Complex Numbers, Unit Circle

  1. February 27th 2010, 08:12 PM #1
    sashikanth
    sashikanth is offline
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    Complex Numbers, Unit Circle

    <br /> \ Let \ z_{1}, \ z_{2}, \ z_{3}, \ ... \ z_{n}, \mbox{ be n complex numbers on the unit circle} \mid z \mid = 1. <br />
    Prove that there exists a number z on the unit circle such that
    <br /> \mid z-z_{1} \mid \mid z-z_{2} \mid ... \mid z-z_{n} \mid \geq 1<br />


    I used geometry and I'm left to prove that
    <br /> \ 2^n \sin(\frac{\theta - \theta_{1}}{2}) \sin(\frac{\theta - \theta_{2}}{2}) ... \sin(\frac{\theta - \theta_{n}}{2}) \geq 1<br /> \mbox{ where } \theta_{i} \mbox{ is the argument of } \ z_{i}<br />
    I'm stuck here. Is there any other approach to this problem? Please help!
    Last edited by sashikanth; February 27th 2010 at 08:52 PM.
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