i would go that way:
we have 3 points A, B, C with spheres around them with radiuses r_a, r_b and r_c. a point P has to be on all 3 sphere surfaces to be an intersection point.
so how can we formulate that mathematically?
lets start with a try of describing the relationship of a point P on the surface of a sphere to the center A and radius r_a.
one solution would be to state that the radius r_a is just the distance from A to P (the point on the surface):
length (P - A) = r_a
the same is true for the other spheres (with same point P)
length (P - B) = r_b
length (P - C) = r_c
so we have 3 equations and 3 open variables P.x, P.y and P.z.
so it should be solvable that way.
length(d) = sqrt( d.xd.x + d.yd.y + d.z*d.z )
if we know that the radius r_a is positive we can skip the sqrt by sqr-ing the equation
d_a.xd_a.x + d_a.yd_a.y + d_a.zd_a.z = r_ar_a
with d_a = P - A
now it would be good to use a math program to solve the equations for P.x, P.y and P.z.