### 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?

### Pericut

Two semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?

### OK! Now Prove It

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

# Thebault's Theorem

### Why dothis problem?

It gives an opportunity for experimentation and making your own conjecture, which then needs to be proved. Three different methods of proof are based on coordinates, vectors and complex numbers.

### Key questions

Four points are vertices of a parallelogram. What do you know about the line segments joining them?

If on a line segment you draw a square:
• what do you know about the vertices of the square?
• can you find the coordinates, or the position vector of the centre of the square?

### Possible support

The problem Napolean's Theorem is similar and perhaps easier.