Prove the Two Circle Theorem: If circles O and O' having radii r ≥ r', respectivel...

Prove the Two Circle Theorem: If circles O and O' having radii r ≥ r', respectively, have their centers at a distance d apart, where r - r'<d<r + r', then the circles will meet at two distinct points. (Hint: First define for any circle O' and diameter  the function d(x) = OP where O is any fixed point on line  and P varies on circle O' such that x =  Prove that d(x) is continuous using an argument similar to that of Problem 19, then use the given inequalities and the Intermediate Value Theorem to prove there exists point P on circle O' such that d(x) = OP = r, hence P lies on both circles.)