I am working on the problem of finding "rational" dodecahedrons, 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.

Edit: Given that the general problem has in some sense been solved, I would like to focus on a particular set of coordinates. The specific matrix I want to work with is:

\begin{pmatrix}(0,-y)&(0,y)&(x,0) \\ (0,y)&(-a,b)&(a,b) \\ (x,0)&(a,b)&(\frac{a^2+b^2}x,0)\end{pmatrix}

I am hoping to arrive at a diophantine equation in a, b, x, and y of small degree that would allow a full characterization (or perhaps an infinite family) of rational dodecahedron.

One sanity check that this actually works: a=22, b=54, x=40, y=10 should be a solution (because it does generate a rational dodecahedron). I tried working this out myself but I keep not getting it equal to zero.