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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?

Sliding Coins

Stage: 3 and 4 Short Challenge Level: Challenge Level:1
We cannot remove two touching coins; if you did, any third coin originally making a triangle with then would then be slideable. We are able to remove the centre and corner coins without enabling any of the remaining coins to slide.If we do this, all of the coins touched at least one of the removed coins so we cannot remove any more.

If we remove a coin which is not one of these four coins first, it would be one of a middle pair of an edge. Four coins touch this coin, leaving five possible coins to remvoe. Four of the five coins would lie along an edge of the frame, with two of these forming a triangle with the fifth. Clearly, only one of the three coins which form a triangle may now be removed, together with only one of the other two coins as they are touching. So we conclude that the maximum number of coins which may be removed is four.

This problem is taken from the UKMT Mathematical Challenges.