Polynomial system to compute all circles that are tangent to 3 given conics. See https://www.JuliaHomotopyContinuation.org/blog/circles_conics/

circles_conic.jl

x, y, a_1, a_2, r, b_1, b_2, b_3, b_4, b_5, b_6 = [PolyVar{true}(i) for i in ["x", "y", "a_1", "a_2", "r", "b_1", "b_2", "b_3", "b_4", "b_5", "b_6"]];
f = [a_1^8*b_1^2*b_2^4+2*a_1^6*a_2^2*b_1^2*b_2^4+a_1^4*a_2^4*b_1^2*b_2^4-4*a_1^6*r^2*b_1^2*b_2^4-6*a_1^4*a_2^2*r^2*b_1^2*b_2^4-2*a_1^2*a_2^4*r^2*b_1^2*b_2^4+6*a_1^4*r^4*b_1^2*b_2^4+6*a_1^2*a_2^2*r^4*b_1^2*b_2^4+a_2^4*r^4*b_1^2*b_2^4-4*a_1^2*r^6*b_1^2*b_2^4-2*a_2^2*r^6*b_1^2*b_2^4+r^8*b_1^2*b_2^4+2*a_1^7*a_2*b_1*b_2^5+4*a_1^5*a_2^3*b_1*b_2^5+2*a_1^3*a_2^5*b_1*b_2^5-6*a_1^5*a_2*r^2*b_1*b_2^5-8*a_1^3*a_2^3*r^2*b_1*b_2^5-2*a_1*a_2^5*r^2*b_1*b_2^5+6*a_1^3*a_2*r^4*b_1*b_2^5+4*a_1*a_2^3*r^4*b_1*b_2^5-2*a_1*a_2*r^6*b_1*b_2^5+a_1^6*a_2^2*b_2^6+2*a_1^4*a_2^4*b_2^6+a_1^2*a_2^6*b_2^6-a_1^6*r^2*b_2^6-5*a_1^4*a_2^2*r^2*b_2^6-5*a_1^2*a_2^4*r^2*b_2^6-a_2^6*r^2*b_2^6+3*a_1^4*r^4*b_2^6+7*a_1^2*a_2^2*r^4*b_2^6+3*a_2^4*r^4*b_2^6-3*a_1^2*r^6*b_2^6-3*a_2^2*r^6*b_2^6+r^8*b_2^6-8*a_1^8*b_1^3*b_2^2*b_3-16*a_1^6*a_2^2*b_1^3*b_2^2*b_3-8*a_1^4*a_2^4*b_1^3*b_2^2*b_3+32*a_1^6*r^2*b_1^3*b_2^2*b_3+48*a_1^4*a_2^2*r^2*b_1^3*b_2^2*b_3+16*a_1^2*a_2^4*r^2*b_1^3*b_2^2*b_3-48*a_1^4*r^4*b_1^3*b_2^2*b_3-48*a_1^2*a_2^2*r^4*b_1^3*b_2^2*b_3-8*a_2^4*r^4*b_1^3*b_2^2*b_3+32*a_1^2*r^6*b_1^3*b_2^2*b_3+16*a_2^2*r^6*b_1^3*b_2^2*b_3-8*r^8*b_1^3*b_2^2*b_3-16*a_1^7*a_2*b_1^2*b_2^3*b_3-32*a_1^5*a_2^3*b_1^2*b_2^3*b_3-16*a_1^3*a_2^5*b_1^2*b_2^3*b_3+48*a_1^5*a_2*r^2*b_1^2*b_2^3*b_3+64*a_1^3*a_2^3*r^2*b_1^2*b_2^3*b_3+16*a_1*a_2^5*r^2*b_1^2*b_2^3*b_3-48*a_1^3*a_2*r^4*b_1^2*b_2^3*b_3-32*a_1*a_2^3*r^4*b_1^2*b_2^3*b_3+16*a_1*a_2*r^6*b_1^2*b_2^3*b_3-6*a_1^6*a_2^2*b_1*b_2^4*b_3-12*a_1^4*a_2^4*b_1*b_2^4*b_3-6*a_1^2*a_2^6*b_1*b_2^4*b_3+10*a_1^6*r^2*b_1*b_2^4*b_3+42*a_1^4*a_2^2*r^2*b_1*b_2^4*b_3+42*a_1^2*a_2^4*r^2*b_1*b_2^4*b_3+10*a_2^6*r^2*b_1*b_2^4*b_3-30*a_1^4*r^4*b_1*b_2^4*b_3-66*a_1^2*a_2^2*r^4*b_1*b_2^4*b_3-30*a_2^4*r^4*b_1*b_2^4*b_3+30*a_1^2*r^6*b_1*b_2^4*b_3+30*a_2^2*r^6*b_1*b_2^4*b_3-10*r^8*b_1*b_2^4*b_3+2*a_1^5*a_2^3*b_2^5*b_3+4*a_1^3*a_2^5*b_2^5*b_3+2*a_1*a_2^7*b_2^5*b_3-2*a_1^5*a_2*r^2*b_2^5*b_3-8*a_1^3*a_2^3*r^2*b_2^5*b_3-6*a_1*a_2^5*r^2*b_2^5*b_3+4*a_1^3*a_2*r^4*b_2^5*b_3+6*a_1*a_2^3*r^4*b_2^5*b_3-2*a_1*a_2*r^6*b_2^5*b_3+16*a_1^8*b_1^4*b_3^2+32*a_1^6*a_2^2*b_1^4*b_3^2+16*a_1^4*a_2^4*b_1^4*b_3^2-64*a_1^6*r^2*b_1^4*b_3^2-96*a_1^4*a_2^2*r^2*b_1^4*b_3^2-32*a_1^2*a_2^4*r^2*b_1^4*b_3^2+96*a_1^4*r^4*b_1^4*b_3^2+96*a_1^2*a_2^2*r^4*b_1^4*b_3^2+16*a_2^4*r^4*b_1^4*b_3^2-64*a_1^2*r^6*b_1^4*b_3^2-32*a_2^2*r^6*b_1^4*b_3^2+16*r^8*b_1^4*b_3^2+32*a_1^7*a_2*b_1^3*b_2*b_3^2+64*a_1^5*a_2^3*b_1^3*b_2*b_3^2+32*a_1^3*a_2^5*b_1^3*b_2*b_3^2-96*a_1^5*a_2*r^2*b_1^3*b_2*b_3^2-128*a_1^3*a_2^3*r^2*b_1^3*b_2*b_3^2-32*a_1*a_2^5*r^2*b_1^3*b_2*b_3^2+96*a_1^3*a_2*r^4*b_1^3*b_2*b_3^2+64*a_1*a_2^3*r^4*b_1^3*b_2*b_3^2-32*a_1*a_2*r^6*b_1^3*b_2*b_3^2-32*a_1^6*r^2*b_1^2*b_2^2*b_3^2-96*a_1^4*a_2^2*r^2*b_1^2*b_2^2*b_3^2-96*a_1^2*a_2^4*r^2*b_1^2*b_2^2*b_3^2-32*a_2^6*r^2*b_1^2*b_2^2*b_3^2+96*a_1^4*r^4*b_1^2*b_2^2*b_3^2+192*a_1^2*a_2^2*r^4*b_1^2*b_2^2*b_3^2+96*a_2^4*r^4*b_1^2*b_2^2*b_3^2-96*a_1^2*r^6*b_1^2*b_2^2*b_3^2-96*a_2^2*r^6*b_1^2*b_2^2*b_3^2+32*r^8*b_1^2*b_2^2*b_3^2-16*a_1^5*a_2^3*b_1*b_2^3*b_3^2-32*a_1^3*a_2^5*b_1*b_2^3*b_3^2-16*a_1*a_2^7*b_1*b_2^3*b_3^2+16*a_1^5*a_2*r^2*b_1*b_2^3*b_3^2+64*a_1^3*a_2^3*r^2*b_1*b_2^3*b_3^2+48*a_1*a_2^5*r^2*b_1*b_2^3*b_3^2-32*a_1^3*a_2*r^4*b_1*b_2^3*b_3^2-48*a_1*a_2^3*r^4*b_1*b_2^3*b_3^2+16*a_1*a_2*r^6*b_1*b_2^3*b_3^2+a_1^4*a_2^4*b_2^4*b_3^2+2*a_1^2*a_2^6*b_2^4*b_3^2+a_2^8*b_2^4*b_3^2-2*a_1^4*a_2^2*r^2*b_2^4*b_3^2-6*a_1^2*a_2^4*r^2*b_2^4*b_3^2-4*a_2^6*r^2*b_2^4*b_3^2+a_1^4*r^4*b_2^4*b_3^2+6*a_1^2*a_2^2*r^4*b_2^4*b_3^2+6*a_2^4*r^4*b_2^4*b_3^2-2*a_1^2*r^6*b_2^4*b_3^2-4*a_2^2*r^6*b_2^4*b_3^2+r^8*b_2^4*b_3^2+32*a_1^6*a_2^2*b_1^3*b_3^3+64*a_1^4*a_2^4*b_1^3*b_3^3+32*a_1^2*a_2^6*b_1^3*b_3^3+32*a_1^6*r^2*b_1^3*b_3^3+32*a_1^4*a_2^2*r^2*b_1^3*b_3^3+32*a_1^2*a_2^4*r^2*b_1^3*b_3^3+32*a_2^6*r^2*b_1^3*b_3^3-96*a_1^4*r^4*b_1^3*b_3^3-160*a_1^2*a_2^2*r^4*b_1^3*b_3^3-96*a_2^4*r^4*b_1^3*b_3^3+96*a_1^2*r^6*b_1^3*b_3^3+96*a_2^2*r^6*b_1^3*b_3^3-32*r^8*b_1^3*b_3^3+32*a_1^5*a_2^3*b_1^2*b_2*b_3^3+64*a_1^3*a_2^5*b_1^2*b_2*b_3^3+32*a_1*a_2^7*b_1^2*b_2*b_3^3-32*a_1^5*a_2*r^2*b_1^2*b_2*b_3^3-128*a_1^3*a_2^3*r^2*b_1^2*b_2*b_3^3-96*a_1*a_2^5*r^2*b_1^2*b_2*b_3^3+64*a_1^3*a_2*r^4*b_1^2*b_2*b_3^3+96*a_1*a_2^3*r^4*b_1^2*b_2*b_3^3-32*a_1*a_2*r^6*b_1^2*b_2*b_3^3-8*a_1^4*a_2^4*b_1*b_2^2*b_3^3-16*a_1^2*a_2^6*b_1*b_2^2*b_3^3-8*a_2^8*b_1*b_2^2*b_3^3+16*a_1^4*a_2^2*r^2*b_1*b_2^2*b_3^3+48*a_1^2*a_2^4*r^2*b_1*b_2^2*b_3^3+32*a_2^6*r^2*b_1*b_2^2*b_3^3-8*a_1^4*r^4*b_1*b_2^2*b_3^3-48*a_1^2*a_2^2*r^4*b_1*b_2^2*b_3^3-48*a_2^4*r^4*b_1*b_2^2*b_3^3+16*a_1^2*r^6*b_1*b_2^2*b_3^3+32*a_2^2*r^6*b_1*b_2^2*b_3^3-8*r^8*b_1*b_2^2*b_3^3+16*a_1^4*a_2^4*b_1^2*b_3^4+32*a_1^2*a_2^6*b_1^2*b_3^4+16*a_2^8*b_1^2*b_3^4-32*a_1^4*a_2^2*r^2*b_1^2*b_3^4-96*a_1^2*a_2^4*r^2*b_1^2*b_3^4-64*a_2^6*r^2*b_1^2*b_3^4+16*a_1^4*r^4*b_1^2*b_3^4+96*a_1^2*a_2^2*r^4*b_1^2*b_3^4+96*a_2^4*r^4*b_1^2*b_3^4-32*a_1^2*r^6*b_1^2*b_3^4-64*a_2^2*r^6*b_1^2*b_3^4+16*r^8*b_1^2*b_3^4+4*a_1^6*a_2*b_1^2*b_2^3*b_4+4*a_1^4*a_2^3*b_1^2*b_2^3*b_4-12*a_1^4*a_2*r^2*b_1^2*b_2^3*b_4-8*a_1^2*a_2^3*r^2*b_1^2*b_2^3*b_4+12*a_1^2*a_2*r^4*b_1^2*b_2^3*b_4+4*a_2^3*r^4*b_1^2*b_2^3*b_4-4*a_2*r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