If three circles intersect at one point then there's unique $x$ and $y$ coordinate values such that the following equations are satisfied:
$$(x-x_i)^2 + (y-y_i)^2 = r_i^2$$
Where $i=1,2,3$
Taking difference of consecutive pairs would give me 3 linear equations with 2 unknowns which I can solve for the unknowns.
Is that the correct way to find the $x$ and $y$ values?
I'd not say that this is the correct way to find the point of intersection, but it certainly is one correct and reasonable way.