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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:

This problem builds on from Points in Pairs - 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 along the dotted line.

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

If P' moves along a straight line what does P do?


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