If one the vertices of the square circumscribing the circle $$\left| {z - 1} \right| = \sqrt 2 \,is\,2 + \sqrt {3\,} \,i$$. Find the other vertices of the square.

Find the centre and radius of circle given by $$\,\left| {{{z - \alpha } \over {z - \beta }}} \right| = k,k \ne 1\,$$

where, $${\rm{z = x + iy, }}\alpha {\rm{ = }}\,{\alpha _1}{\rm{ + i}}{\alpha _2}{\rm{,}}\,\beta = {\beta _1}{\rm{ + i}}{\beta _2}{\rm{ }}$$

If $${z_1}$$ and $${z_2}$$ are two complex numbers such that $$\,\left| {{z_1}} \right| < 1 < \left| {{z_2}} \right|\,$$ then prove that $$\,\left| {{{1 - {z_1}\overline {{z_2}} } \over {{z_1} - {z_2}}}} \right| < 1$$.

Prove that there exists no complex number z such that $$\left| z \right| < {1 \over 3}\,and\,\sum\limits_{r = 1}^n {{a_r}{z^r}} = 1$$ where $$\left| {{a_r}} \right| < 2$$.