A Ford circle 
is a circle of radius 
centered at 
, where 
and 
are relatively prime integers
and 
. Distinct Ford circles have disjoint
interiors and may be tangent.
All are tangent to the x-axis.
This can be seen by examining the squared distance between
the centers of two distinct circles with 
and 
,
 |
(1)
|
Let 
be the sum of the radii
 |
(2)
|
then
 |
(3)
|
But 
, so 
and the distance
between circle centers is 
the sum of the circle radii,
with equality (and therefore tangency) iff 
. Ford circles are related
to the Farey sequence (Conway and Guy 1996).
If 
, 
, and 
are three consecutive terms in a Farey
sequence, then the circles 
and 
are tangent at
 |
(4)
|
and the circles 
and 
are tangent at
 |
(5)
|
Moreover, 
lies on the circumference of the semicircle
with diameter 
and 
lies on the circumference
of the semicircle with diameter 
(Apostol 1997,
p. 101).
