I am working on the problem of finding ["rational" dodecahedrons](https://mathoverflow.net/questions/234212/rational-inscribed-realization-of-the-regular-dodecahedron), and I have run across an interesting subproblem: How do you tell if three circles have a common intersection point? 

Specifically, given the points: $P_{ij}=(x_{ij},y_{ij}),  1\leq i,j \leq 3$, such that the circles $C_i$ are determined by $P_{ij}$, is there a formula based on the points $P_{ij}$ to determine whether the circles $C_i$ all intersect at one common point? If it helps, for the problem in question one may take $P_{ij}=P_{ji}$.

What would be particularly pleasing is if the formula could be represented as a determinant of a matrix (or potentially a determinant like object for a tensor). This would be reminiscent of the determinant condition for four points to be concyclic.