Pick any two integers h and k, then the circle C(h,k) of radius centered at is known as a Ford circle. No matter what and how many hs and ks are picked, none of the Ford circles intersect (and all are tangent to the x-axis). This can be seen by examining the squared distance between the centers of the circles with (h, k) and ,

Let s be the sum of the radii

then

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

and the circles and intersect in

Moreover, lies on the circumference of the semicircle with diameter and lies on the circumference of the semicircle with diameter (Apostol 1997, p. 101).

 

Adjacent Fraction, Apollonian Gasket, Farey Sequence, Stern-Brocot Tree