### 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?

# Mapping the Wandering Circle

##### Age 14 to 16 Challenge Level:

This problem builds on from The Line and Its Strange Pair - you may wish to look at that first.

Two points, one inside a circle and the other outside, are related in the following way :

A line starting at the centre of the circle and passing through the first point ( P ) goes on to pass through the second point ( P' )

Positions along the line are such that the ratio of OP to the radius of the circle matches the ratio of the radius of the circle to OP'

For example if OP happened to be 2/3 of the radius then OP' would be 3/2 of the radius.

In the diagram above, the point P can move to different places around the dotted circle.

Each position P takes will fix a corresponding position for P' .

As P moves around that circle what will P' do ?

Why?

You might find the interactivity below helpful. Holding down the mouse button leaves a trace of the points.

Full Screen Version