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# Ford Circles

### Why do this problem?

This problem provides a beautiful extension to Farey Neighbours and encourages students to explore the connection between a geometrical pattern and a numerical sequence.

### Possible approach

It would be helpful if students were familiar with Farey Sequences and Farey Neighbours.
Start by demonstrating the first GeoGebra applet and defining 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)."

Invite students to explore the second GeoGebra applet and challenge them to find some values of $a, b, c$ and $d$ that generate circles which touch. Record any that they find, and invite them to look for patterns, drawing attention to $ad-bc$ if it does not emerge from the class.

The two questions at the end of the problem are the key to the link between Ford Circles and Farey Sequences, and provide a good challenge in proof for older students:

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

A diagram like this one might help: $R=\frac{1}{2d^2}$ and $r=\frac{1}{2c^2}$.
The centre of the circle which touches the horizontal axis at $M$ is $(\frac{a+b}{c+d}, \frac{1}{2(c+d)^2})$.

### Key questions

What can you say about $R+r$ and $R-r$ if the circles centre $B$ and $A$ just touch each other?
Can you use Pythagoras theorem?

### Possible extension

Students may wish to read more about Ford Circles in this Wikipedia article.

### Possible support

As well as working on Farey NeighboursBaby Circle would be a useful problem to try before attempting this challenging task.