The Arbelos
If three semicircles R, S and T
touch at A, B and C,
This result is called the arbelos since the shape bounded by the semicircles


Proof As usual, we invert the picture in a wellchosen circle.
Let L be the circle centre C, passing through A.
Now it is clear that we can draw D_{1} touching
It is easy to see that we can add a further circle, D_{2},
When we invert these in L, we get the required C_{n}.


The CabriJava plane allows you to experiment. The first five circles of the chain are shown. Drag A, B or C to change the picture. A web page just for the arbelos

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