3. This proof isn't entirely rigorous, but contains all the major steps:
Let D be the point where and are tangent. Now let us define a Möbius transform that maps D to infinity, and let us indicate images under M by primes. Now, M maps and to parallel lines and . Thus are circles tangent to the lines and and to the adjacent circles in the sequence. As they are tangent to both and , they are all the same size, and their centers lie on the line parallel to and and halfway between them. We see also that must lie on as well. Thus if we have the circle in the pre-image, we see all lie on .