Explanation

(a)
We have, $\mathcal{|z -}\mathcal{z}_{1}| = \left| \mathcal{z -}\mathcal{z}_{2} \right| = \left| \mathcal{z -}\mathcal{z}_{3} \right|\mathcal{= |z -}\mathcal{z}_{4}|$
Therefore, the point having affix $\mathcal{z}$ is equidistant from the four points having affixes $\mathcal{z}_{1},\ \mathcal{z}_{2},\ \mathcal{z}_{3},\ \mathcal{z}_{4}$. Thus $\mathcal{z}$ is the affix of either the centre of a circle or the point of intersection of diagonals of a square (or rectangle). Therefore, $\mathcal{z}_{1},\mathcal{z}_{2},\ \mathcal{z}_{3},\mathcal{z}_{4}$ are either concyclic or vertices of a square (of rectangle). Hence, $\mathcal{z}_{1},\mathcal{z}_{2},\ \mathcal{z}_{3},\mathcal{z}_{4}$ are concyclic