Ford Circle

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 ,

 (1)

Let s 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 intersect in

 (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).

Apostol, T. M. "Ford Circles." §5.5 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 99-102, 1997.

Conway, J. H. and Guy, R. K. "Farey Fractions and Ford Circles." The Book of Numbers. New York: Springer-Verlag, pp. 152-154, 1996.

Ford, L. R. "Fractions." Amer. Math. Monthly 45, 586-601, 1938.

Pickover, C. A. "Fractal Milkshakes and Infinite Archery." Ch. 14 in Keys to Infinity. New York: W. H. Freeman, pp. 117-125, 1995.

Rademacher, H. Higher Mathematics from an Elementary Point of View. Boston, MA: Birkhäuser, 1983.