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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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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?

Three Four Five

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

3 semicircles
As shown in the diagram, two semi-circles (each of radius $1/2$) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles, and find a 3-4-5 triangle in the diagram.