Figure 15: Notation for the elementary geometric lemma.
Lemma.
Let 
, 
, 
be three circles bounding disjoint
open disks in the plane.
(See Figure 15.)
Let the radii of these circles be 
, 
, 
.
Assume that
Let 
and 
denote the points on
closest to and .
Let
and
Then for universal constants 
, 
,
if
then
This proposition remains true in the limit when
one or both of the circles and
are allowed to degenerate into lines
(circles of infinite radius).
Proof.
Choose any between 0 and 1.
Call a configuration of circles a -configuration if
Clearly
is a continuous positive function
on the space of -configurations.
The only way for
to approach 0 is for something bad to happen
as one of the radii
approaches 0,
but it's easy to see that no such bad thing happens.
Hence there is some appropriate choice for .
More concretely,
consider the two special -configurations for which
,
,
,
and either
or
.
(See Figure 16.)
Figure 16: Special configurations
An arbitrary -configuration
can be transformed into
one of these two special configurations
by a sequence of steps that do not increase
.
(Without going into detail,
the steps are:
make 
;
assume 
;
make ;
make ;
either make
or make
;
make 
.)
Setting
and computing
for the two special configurations,
we find
(see Figure 17)
that we can choose
.