Ford circle theorem

In mathematics, a Ford circle is a circle in the Euclidean plane, in a family of circles that are all tangent to the $x$-axis at rational points. 

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For each rational number $\dfrac{p}{q}$, expressed in lowest terms, there is a Ford circle whose center is at the point $(\dfrac{p}{q}, \dfrac{1}{2q^2})$ and whose radius is $\dfrac{1}{2q^2}$. It is tangent to the x-axis at its bottom point, $(\dfrac{p}{q},0)$. 

The two Ford circles for rational numbers $\dfrac{p}{q}$ and $\dfrac{r}{s}$ (both in lowest terms) are tangent circles when $|ps−qr|=1$|and otherwise these two circles are disjoint.

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