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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?

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Polycircles

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?

Quad in Quad

Stage: 4 Challenge Level: Challenge Level:1
ABCD is a convex quadrilateral. The points P, Q, R and S are the midpoints of the edges of ABCD. You can change the shape of the quadrilateral ABCD.

What do you notice about the quadrilateral PQRS as ABCD changes?

Is the area of PQRS always the same fraction of the area of ABCD and, if so, what is this fraction?

Try to prove your conjectures.

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