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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?

Triangles Within Squares

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2
Why not encourage pupils to discover rules of their own?

This problem links to "Triangles withinTriangles " and the problem "Triangles within Pentagons"

There are many different ways to visualise this question and pupils should be encouraged to explain how they "know" their rule works.