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)&(-c,d)&(c,d) \\ (x,0)&(c,d)&(\frac{c^2+d^2}x,0)\end{pmatrix}

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

Edit 2: So I have used Gro-Tsen's degree six equation, and plugged in the points from the dodecahedron matrix. The resulting equation is of degree 8, and after removing the trivial factors $cy(c^2+d^2-x^2)$, the final equation is: $cxy^2-d^2y^2-c^2y^2+dx^2y+d^3y+c^2dy-d^2x^2-c^2x^2+cd^2x+c^3x=0$. You can rewrite this more simply as $xy(cy+dx)+(c^2+d^2)(cx+dy-x^2-y^2)=0$. This degree 4 Diophantine equation is the condition for a rational dodecahedron.

One final thing I would like to know... Is Gro-Tsen's formula for the symmetric 3x3 set of coordinates actually a determinant of a 6x6 matrix? It seems likely, given that it is degree six and has 720 terms (of which exactly half are negative). If so, what is that matrix?