FordŐs Touching Circles

For the pictorial presentation we begin with a straight line that can be considered the x-axis in the plane.  For any rational number , a circle tangent to that point with a diameter  can be drawn.  Some circles are tangent and others are not.

As discovered by Lester R. Ford, if two circles corresponding to fractions  and  touch, then the circle corresponding to the mediant fraction, , is tangent to them both.

Recall, in the Farey Series, the middle of any three successive terms is the mediant of the other two.  Therefore, the Farey Series  can be interpreted geometrically using FordŐs Touching Circles.

To prove this idea that if two circle corresponding to fractions  and  touch, then the circle corresponding to the mediant fraction  is tangent to both of them we need the following theorem.

 The two circles corresponding to  and  touch if and only if ad and bc are consecutive integers.

For a complete proof several cases of the relationship between the integers involved must be considered.  We will consider the following case where b<d and  < .

By construction,

when plugging the above determined values for AB, BC and AC into the Pythagorean Theorem it directly follows that

.

Thus it follows that the two circles are tangent.

From , we can say  and , thus the mediant circle touches the other two if and only if they are tangent to each other.