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Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

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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?

Roll On

Stage: 4 Short Challenge Level: Challenge Level:1

$3$ times

In returning to its original position, the centre of the disc moves around the circumference of a circle of diameter $3d$, i. e. a distance $3\pi d$. As the disc does not slip, its centre moves a distance $\pi d$ when the disc makes one complete turn about its centre, so $3$ complete turns are made.
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.  

This problem is taken from the UKMT Mathematical Challenges.
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