If from the initial circles with curvature $k_a, k_b, k_c, k_d$ and centers $c_a, c_b, c_c, c_d$ you generate four new circles with curvatures:
$ k_e = 2(b+c+d) - a \\ k_f = 2(c+d+a) - b \\ k_g = 2(d+a+b) - c \\ k_h = 2(a+b+c) - d \\ $
Those circles have centers (complex plane):
$ c_e = 2(k_bc_b +k_cc_c +k_dc_d) - k_ac_a $