If $$i{z^3} + {z^2} - z + i = 0$$ , then show that $$\left| z \right| = 1$$.

If $$\left| {Z - W} \right| \le 1,\left| W \right| \le 1$$, show that $${\left| {Z - W} \right|^2} \le {(\left| Z \right| - \left| W \right|)^2} + {(ArgZ - Arg\,W)^2}$$

Let $${z_1}$$ = 10 + 6i and $${z_2}$$ = 4 + 6i. If Z is any complex number such that the argument of $${{(z - {z_1})} \over {(z - {z_2})}}\,is{\pi \over 4}$$ , then prove that $$\left| {z - 7 - 9i} \right| = 3\sqrt 2 $$.

Show that the area of the triangle on the Argand diagram formed by the complex numbers z, iz and z + iz is $${1 \over 2}\,{\left| z \right|^2}$$ .