My attempt to find the equation of circle For $4$ given complex points $z_1, z_2, z_3, z_4$ to be concyclic The complex number given by $$\frac{(z_3-z_1)(z_4-z_2)}{(z_3-z_2)(z_4-z_1)}\ \text{must be real}$$
I replaced one from $z_1, z_2, z_3, z_4$ by variable $z$. Then condition for these points to be concyclic (so that locus of point $z$ would give a circle in complex plane) is
The complex number $$\frac{(z-z_2)(z_3-z_1)}{(z-z_1)(z_3-z_2)}\ \text{must be purely real}$$
I don't know what to do next.
In my book final answer is given as
$$\frac{(z-z_2)(z_3-z_1)}{(z-z_1)(z_3-z_2)} = \frac{(\overline{z}-\overline{z}_2)(\overline{z}_3-\overline{z}_1)}{(\overline{z}-\overline{z}_1)(\overline{z}_3-\overline{z}_2)}$$