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P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

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A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square?

Tilting Triangles

Age 14 to 16 Challenge Level:

As the triangle rotates compare what part of the square is not covered by it anymore and what part of the square is newly covered.

Does the triangle continue to cover the whole of the portion of the square it moves over as its size reduces. What is special when the triangles side is of length $ \sqrt 2 $ units?

Is there a point where the whole of the triangle is always in the square and what effect does that have?