I'm trying to solve the following problem algorithmically. It seems it's a bit too math-y for StackOverflow.
Suppose we're given the coordinates of the centers of three circles ($A,B$ and $C$), as well as the radius of each circle. Now, suppose we iteratively add some small amount to the radius of each circle, until all three circles intersect at some point, $[x_i,y_i]$. How would I go about determining their point of intersection? Now, of course, in practice I can only increase the radius by some discrete amount (maybe $0.01$ units), so I'll have to take into account that they may never actually intersect, rather I'll probably want to determine when they're within some threshold (unless there's a better way to accomplish this entirely). With each iteration, I'll also have to be able to determine whether such an intersection even exists.
The best idea I can think of is to somehow compute every possible point along the circumference of each circle (in practice, a large discrete number, of course), and then check to see if there are three points within some threshold (i.e. there exists 3 points, one contributed by each circle, within some small threshold, maybe $0.01$ units).
Most important here is efficiency and ease of implementation.
NOTE: I should've added-- I'm aware of the solution involving finding the intersection of two hyperbolas. However, for this exercise, I'd like to avoid that particular solution.