### Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

### 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?

### 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?

# Triangles Within Squares

##### Stage: 4 Challenge Level:

The diagram above shows that: $$8 \times T_2 + 1 = 25 = 5^2$$

Use a similar method to help you verify that: $$8 \times T_3 + 1 = 49 = 7^2$$ Can you generalise this result?

Can you find a rule in terms of $T_n$ and a related square number?

Can you find a similar rule involving square numbers for $T_{n}, T_{n+2}$ and several copies of $T_{n+1}$?