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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 Pentagons

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Having identified the rule is it possible to work backwards and reconstruct pentagons using the triangular numbers?

This problem builds on the work of the three problems "sequences and series ", "triangles within triangles " and "triangles within squares ".

There are many more patterns and relationships involving triangular numbers and some experimentation could result in some pupil generated discoveries.