http://math.stackexchange.com/questions/256100/how-can-i-find-the-points-at-which-two-circles-intersect

Two circles with the center points (x1,y1), (x2,y2) and the radii r1 and r2 respective.
(x−x1)^2+(y−y1)^2=r1^2
(x−x2)^2+(y−y2)^2=r2^2

Multiply powers
(x^2 - 2*x*x1 + x1^2) + (y^2 - 2*y*y1 + y1^2) = r1^2
(x^2 - 2*x*x2 + x2^2) + (y^2 - 2*y*y2 + y2^2) = r2^2


Let k*r1 = r2

Subsitute r2 by k*r1
(x^2 - 2*x*x1 + x1^2) + (y^2 - 2*y*y1 + y1^2) = r1^2
(x^2 - 2*x*x2 + x2^2) + (y^2 - 2*y*y2 + y2^2) = (k*r1)^2

# Divide by k^2 to get the same term on the right side.
# (x^2 - 2*x*x1 + x1^2) + (y^2 - 2*y*y1 + y1^2)       = r1^2
# (x^2 - 2*x*x2 + x2^2) + (y^2 - 2*y*y2 + y2^2) / k^2 = r1^2


Isolate x
 (x^2 - 2*x*x2 + x2^2)                         =       r1^2 + (y^2 - 2*y*y1 + y1^2)
 x^2                                           =       r1^2 + (y^2 - 2*y*y1 + y1^2) + 2*x*x2 + x2^2
 x                                               SQRT( r1^2 + (y^2 - 2*y*y1 + y1^2) + 2*x*x2 + x2^2 )

Now substitide y in the second circle equation
# (x^2 - 2*x*x2 + x2^2) + (y^2 - 2*y*y2 + y2^2) = r2^2
 
 (     SQRT( r1^2 + (y^2 - 2*y*y1 + y1^2) + 2*x*x2 + x2^2 )^2 -
   2 * SQRT( r1^2 + (y^2 - 2*y*y1 + y1^2) + 2*x*x2 + x2^2 ) * x2 +
   x2^2 
 ) +
 (y^2 - 2*y*y2 + y2^2)                          = r2^2



Ehhhmmmm ...
F**k