### Fixing It

A and B are two fixed points on a circle and RS is a variable diamater. What is the locus of the intersection P of AR and BS?

### OK! Now Prove It

Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?

### Polycircles

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

##### Age 14 to 18Challenge Level

What shape do you think is being made by joining the midpoints of the sides?  Which properties of the shape do we have to establish to prove that it is this partilcar type of shape|?

In the diagram below, the diagonals $AC$ and $BD$ of the original shape have been drawn in.

What can you say about the lines $AC$ and $PS$?

What about lines $AC$ and $QR$?

Can you prove you conjecture?