Let $\displaystyle Z_1,Z_2,Z_3$ be 3 non-zero complex numbers and $\displaystyle Z_1\neq Z_2$ if $\displaystyle \left|\begin{array}{ccc}|Z_1|&|Z_2|&|Z_3|\\|Z_2|&| Z_3|&|Z_1|\\|Z_3|&|Z_1|&|Z_2|\end{array}\right| = 0$,
then prove that:

$\displaystyle \arg \left(\frac{Z_3}{Z_2}\right) = \arg \left(\frac{Z_3 - Z_1}{Z_2 - Z_1}\right)^2$