### 8 Methods for Three by One

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different? Which do you like best?

### Napoleon's Theorem

Triangle ABC has equilateral triangles drawn on its edges. Points P, Q and R are the centres of the equilateral triangles. What can you prove about the triangle PQR?

### Twizzle Wind Up

A loopy exploration of z^2+1=0 (z squared plus one) with an eye on winding numbers. Try not to get dizzy!

# Twizzles Venture Forth

##### Stage: 4 Challenge Level:

Remember that i is the name of the twizzle that has a number arrow equal to the unit arrow rotated through 90 degrees.

In other words i = 1 cis 90 .

The twizzle multiplication rule is neatly expressed by saying 'Multiply the number arrow lengths, and add the angles'.

So $i^2 = i \times i= 1 cis 90 \times 1 cis 90 = 1 cis 180 = -1.$ Now you should have enough information to multiply out (z-i)(z+i) .