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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?

Semicircle Stack

Stage: 4 Short Challenge Level: Challenge Level:1

Three semicircles showing the rectangle and parts of a circle
The shaded area may be divided into a $ 2 \times 1$ rectangle plus a semicircle plus two quarter circles (all of radius $1$). Hence the total area is that of the rectangle plus a circle of radius $1$. Making $2 + \pi$

This problem is taken from the UKMT Mathematical Challenges.