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Baby Circle

A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?

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Two semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?

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Two perpendicular lines are tangential to two identical circles that touch. What is the largest circle that can be placed in between the two lines and the two circles and how would you construct it?

Ford Circles

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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.