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 *h*s and *k*s 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

If

(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).

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

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.