The GeoGebra applet below shows the construction of a sequence of Ford Circles. Ford Circles have centre $\left(\frac{p}{q},\frac1{2q^2}\right)$ and radius $\frac1{2q^2}$, where $\frac{p}{q}$ is a fraction in its simplest form (that is, $p$ and $q$ are coprime integers).

In the interactivity below, you can move the sliders to choose values for $a, b, c$ and $d$. The circles have centres $\left(\frac{a}{c},\frac1{2c^2}\right)$ and $\left(\frac{b}{d},\frac1{2d^2}\right)$, and radii $\frac1{2c^2}$ and $\frac1{2d^2}$.

When the two circles touch, they are coloured in blue.

Explore the interactivity and find some values of $a, b, c$ and $d$ that generate circles that touch each other.

In the problem Farey Neighbours, you are invited to explore the value of $ad-bc$ for two adjacent fractions $\frac bd$ and $\frac ac$ from any Farey Sequence.

Explore the value of $ad-bc$ for the touching circles that you have found.

What do you notice?

Can you prove that for any touching circles in the interactivity above, $|ad-bc|=1$?

Can you prove that, given two such circles which touch the $x$ axis at $\frac bd$ and $\frac ac$, the circle with centre $\left(\frac{a+b}{c+d},\frac1{2(c+d)^2}\right)$ and radius $\frac1{2(c+d)^2}$ is tangent to both circles?

In the interactivity below, you can move the sliders to choose values for $a, b, c$ and $d$. The circles have centres $\left(\frac{a}{c},\frac1{2c^2}\right)$ and $\left(\frac{b}{d},\frac1{2d^2}\right)$, and radii $\frac1{2c^2}$ and $\frac1{2d^2}$.

When the two circles touch, they are coloured in blue.

Explore the interactivity and find some values of $a, b, c$ and $d$ that generate circles that touch each other.

In the problem Farey Neighbours, you are invited to explore the value of $ad-bc$ for two adjacent fractions $\frac bd$ and $\frac ac$ from any Farey Sequence.

Explore the value of $ad-bc$ for the touching circles that you have found.

What do you notice?

Can you prove that for any touching circles in the interactivity above, $|ad-bc|=1$?

Can you prove that, given two such circles which touch the $x$ axis at $\frac bd$ and $\frac ac$, the circle with centre $\left(\frac{a+b}{c+d},\frac1{2(c+d)^2}\right)$ and radius $\frac1{2(c+d)^2}$ is tangent to both circles?