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Computing the centers of Apollonian circle packings

If from the initial circles $a,b,c,d$ with curvatures $k_a, k_b, k_c, k_d$ and centers $c_a, c_b, c_c, c_d$ you generate four new circles with curvatures: reflection of $a$: $k_e = 2(k_b+k_c+k_d) - …