Given two circles C1 and C2 in a fixed plane P. Find all the dilatations of P that map C1 to C2.
The absolute value of the ratio R'/R=SO'/SO. S is the center point of the line OO' centers of the sphere.
What does it mean when it says ALL the dilations?
You would solve this byvsetting the dilation point by setting it to the intersection of the tangents. If you think about it, there are two possible places to dilate: one in between the circles, one behind $C_2$.
I found this image off of the internet (I would draw my own but I'm too lazy :P)
The intersection point of inner tangent lines and the intersection points of the outer tangent lines would be a point where you could scale the circle to match the other. Make sense now?