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

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