# Complex Numbers, Unit Circle

• February 27th 2010, 08:12 PM
sashikanth
Complex Numbers, Unit Circle
$
\ Let \ z_{1}, \ z_{2}, \ z_{3}, \ ... \ z_{n}, \mbox{ be n complex numbers on the unit circle} \mid z \mid = 1.
$

Prove that there exists a number z on the unit circle such that
$
\mid z-z_{1} \mid \mid z-z_{2} \mid ... \mid z-z_{n} \mid \geq 1
$

I used geometry and I'm left to prove that
$
\ 2^n \sin(\frac{\theta - \theta_{1}}{2}) \sin(\frac{\theta - \theta_{2}}{2}) ... \sin(\frac{\theta - \theta_{n}}{2}) \geq 1
\mbox{ where } \theta_{i} \mbox{ is the argument of } \ z_{i}
$

I'm stuck here. Is there any other approach to this problem? Please help! :)