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
.
--Kevin C.