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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?

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?

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?

The Line and Its Strange Pair

Age 14 to 16 Challenge Level:

$a$ is the point on the line such that $Oa$ is perpendicular to the line, and A is its pair. For any point p on the line, it's pair $P$ forms a trianle $OAP$ which is similar to triangle $Oap$, since $$\frac{|OP|}{|OA|} = \frac{1/|Op|}{1/|Oa|} = \frac{|Oa|}{|Op|}$$. Therefore $\angle OPA$ is a right angle, and as p moves along the line, P traces out a cirlce with diameter $OA$.