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
If you can see this message Flash may not be working in your browser
Please see http://nrich.maths.org/techhelp/#flash to enable it.