Assume that there's an intersection point. Let particle A be at that point at and particle B be at the point at . Solve the following equations simultaneously for and :

.... (1)

.... (2)

.... (3)

I suggest solving (1) and (2) simultaneously and then testing the solutions in the third. If you get a solution that works for all three equations, then those values can be used to get the cartesian coordinates of the intersection point.