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This problem provides a beautiful extension to Farey Neighbours and encourages students to explore the connection between a geometrical pattern and a numerical sequence.
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:
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 $(\frac{a+b}{c+d}, \frac{1}{2(c+d)^2})$.
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?
Students may wish to read more about Ford Circles in this Wikipedia article.
As well as working on Farey Neighbours, Baby Circle would be a useful problem to try before attempting this challenging task.
A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?
A triangle PQR, right angled at P, slides on a horizontal floor with Q and R in contact with perpendicular walls. What is the locus of P?
A box of size a cm by b cm by c cm is to be wrapped with a square piece of wrapping paper. Without cutting the paper what is the smallest square this can be?