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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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Pericut

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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Polycircles

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
There is a very strong connection between this problem and the "Sequences and Series" problem.

Can you see the rectangles made from two triangular numbers in the square?

Can you explain why there is always one left over?