Two circles are centered at A and C, with adjustable sizes by moving B and D, respectively. E is the center of a blue circle mutually tangent to the given black circles, and the red dashed path is the set of all such centers. Increasing the slider r moves the center of the larger circle far off to the left. You can move E to see how the blue circles change along the path.
What shape is the original dashed path? What do you know about it?
Move A to be just on top of C. What is the dashed path? What do you know about it?
When r is large and the center of the larger circle C is very far away, what does the red dashed path look like? Why? What do you know about it?