If the coordinates of four concyclic points on the rectangular hyperbola $xy=c^2\text{ are }\left(c{t}_i,c/t_i\right)i=1,2,3,4$ then

Equation of a circle be $x^2+y^2+2gx+2fy+k=0$

If the point $(ct,c/t)$ lies on it, then

$c^2{t}^4+2gc{t}^3+k{t}^2+2fct+c^2=0$

which gives four values of $t\text{ say }{t}_1,t_2,t_3,t_4$ and the

four concyclic points such that $t_1{t}_2{t}_3{t}_4=1$.