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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 Triangles

Stage: 4 Challenge Level: Challenge Level:1
It is usefult to be able to recreate these patterns on square dotty paper.
Is the way the triangles fit together always going to work with two consecutive triangular numbers?

Yuu might find it useful to look at the problem on "Sequences and Series".