From now on we will assume that there are three or more circles.
We can assume that some pair of circles are tangent:
If not, pick one of the circles
(say 
)
and enlarge it until it hits another of the circles.
Call the enlarged circle
,
and cut 
into the two pieces
and
By the cutting law,
Hence it suffices to find a lower bound for
The argument we just went through shows that without loss of generality,
we may assume that and 
are tangent.
Moving the point of tangency to 
and normalizing,
we may assume that and
are the lines y=0 and y=1.
(See Figure 5.)
Figure 5: Normalized configuration.