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Just Rolling Round

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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Roaming Rhombus

We have four rods of equal lengths hinged at their endpoints to form a rhombus ABCD. Keeping AB fixed we allow CD to take all possible positions in the plane. What is the locus (or path) of the point D?

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Triangles and Petals

An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?

Symmetric Trace

Age 14 to 16 Challenge Level:
Pranav from Vardhana School had a good suggestion : copy the image from the NRICH webpage into Paint and then make a copy you can rotate and compare.

Also if you copy into Word, and rotate a copy, you could use one of the drawing tool to trace over one of the curves you want to compare and then move that new line around to see if it fits over the rotated copy.

Trace 2 does not line up with itself upside down. This can be spotted by looking at the lowest point on the line, the curve is sharper than at the highest point on the line, and so when rotated these two can not line up.

Trace 3 on the other hand does line up with itself upside down. This can be spotted by the fact that when you rotate the graph and slide it along it lines up. Another method is to draw a line at the top:

You can now notice that when the wheel has turned 180 degrees, then the rotated diagram is the same as the one above, and so the rotated graph does line up with itself.