Complex Numbers

If three complex numbers are in A.P. then they lie on a circle in the complex plane.

If the complex numbers, ${Z_1},{Z_2}$ and ${Z_3}$ represent the vertics of an equilateral triangle such that
$\left| {{Z_1}} \right| = \left| {{Z_2}} \right| = \left| {{Z_3}} \right|$ then ${Z_1} + {Z_2} + {Z_3} = 0.$

For complex number ${z_1} = {x_1} + i{y_1}$ and ${z_2} = {x_2} + i{y_2},$ we write ${z_1} \cap {z_2},\,\,if\,\,{x_1} \le {x_2}\,\,and\,\,{y_1} \le {y_2}.$
Then for all complex numbers $z\,\,with\,\,1 \cap z,$ we have ${{1 - z} \over {1 + z}} \cap 0.$