Let ¯ z denote the complex conjugate of a complex number z and let...

Given $\overset{¯}{z}\left(1 - i\right) = {z}^{2}\left(1 + i\right)$

So \left|\overset{¯}{z}\right|\left|1 - i\right| = {\left|z\right|}^{2}\left|1 + i\right|$</p><p>$⇒ \left|z\right| = {\left|z\right|}^{2} ⇒ \left|z\right| = 0$or \left|z\right| = 1$

Also, $\overset{¯}{z}\left(1 - i\right) = {z}^{2}\left(1 + i\right) ⇒ \overset{¯}{z}\dfrac{\left(1 - i\right)}{\left(1 + i\right)} = {z}^{2}$

$⇒ - i\overset{¯}{z} = {z}^{2}$

Let $z = x + iy$

$⇒ {x}^{2} - {y}^{2} - \left(2xy\right)i = y - ix$

i.e. ${x}^{2} - {y}^{2} = y$ and $- 2xy = - x$

From $- 2xy = - x$, we get

$x = 0\text{or}y = - \dfrac{1}{2}$

i.e. the points are \left(0, 0\right)$, \left(0, 1\right)$, \left(\dfrac{\sqrt{3}}{2}, - \dfrac{1}{2}\right)$and \left(- \dfrac{\sqrt{3}}{2}, - \dfrac{1}{2}\right)$