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Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger 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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Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

Curvy Areas

Stage: 4 Challenge Level: Challenge Level:1

Take a look at the image below:

Can you see how the image was created?
Try to recreate it using a ruler and compasses.

Here are two images created in a similar way.

Can you work out the proportion of the 3-colour, 4-colour and 5-colour circles which is shaded red?
Can you make any generalisations?
Can you prove your ideas?

What about the proportion which is shaded orange? Yellow? ... 
Can you make any generalisations?
Can you prove your ideas? 

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