Three circles (A, B, and C) are tangentially connected:

Properties:
1) Circle B's radius is twice the radius of Circle A.
2) Circle C is of the exact size that if it rolls around A, or it rolls around B, it will touch A and B on the opposite side at identical points E and F.

What is the radius of Circle C if:
1) C > B
2) B > C > A
3) A > C (C is a maximum)