Why do this problem?

The proof only requires the use of some simple circle geometry and Pythagoras Theorem and it establishes the aesthetically pleasing connection between the number patterns in Farey sequences (see the problem Farey Sequences ) and the patterns of touching circles shown in the animation in this problem.

Possible approach

Use this diagram where $R=\frac{1}{2d^2}$ 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

See the problem Farey Neighbours

Possible support

See the problems Farey Sequences and Baby Circle.