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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?

Circles in Quadrilaterals

Stage: 4 Challenge Level: Challenge Level:1

Preveina from Crest Girls' Academy sent us some pictures to support her reasoning about some of the shapes in this problem:

A circle can be always fitted in a square touching all 4 sides since the sides of a square are all equal. This makes the circle touch each side of the square evenly.
A circle can never be fitted in to a rectangle touching all 4 sides because a rectangle has 2 long sides and 2 short sides. When you're trying to draw a circle that touches all 4 sides in a rectangle it'll turn out to be an oval, since there are 2 long sides.
A circle can never be fitted in to a parallelogram touching all 4 sides because a parallelogram has 2 long sides and 2 short sides just like a rectangle has.
A circle can also be fitted in to a kite touching all 4 sides.
The circle can be sometimes fitted into the trapezium touching all 4 sides depending on the length of the sides. If the non-parallel sides are too far apart, the circle becomes stretched into an oval.