If the circles are given by their centers $O_1,O_2,O_3$ and radii $r_1,r_2,r_3$ respectively, then the condition is that the tetrahedron $A_1A_2A_3A_4$ with edge lengths $A_iA_j=O_iO_j$ and $A_4A_i=r_i$ when $i, j<4$ is planar. In other words, its volume is zero. The squared volume is given by the Caley - Menger determinant. Your problem reduces to this data by finding the centers and radii if circumcircles which is rather standard.

If the circles are given by their centers $O_1,O_2,O_3$ and radii $r_1,r_2,r_3$ respectively, then the condition is that the tetrahedron $A_1A_2A_3A_4$ with edge lengths $A_iA_j=O_iO_j$ and $A_4A_i=r_i$ when $i, j<4$ is planar. In other words, its volume is zero. The squared volume is given by the Caley - Menger determinant. Your problem reduces to this data by finding the centers and radii of circumcircles which is rather standard.