### 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 $OA$.