### 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 Pentagons

##### Stage: 4 Challenge Level:

The diagram above shows the pentagon growing in a systematic way to produce the pentagonal numbers. The equations below describe this growth:
$$\begin{eqnarray} P_2 &=& P_1 + 4 \\ &=&1+4 \\ &=&1 + 3 \times 1 + 1\\ P_3&=& P_2 + 7 \\ &=& 1+4+7 \\ &=& 1 + (3 \times 1 + 1) + (3 \times 2 + 1) \\ P_4 &=& P_3 + 10 \\ &=& 1 + (3 \times 1 + 1) + (3 \times 2 + 1) + (3 \times 3 + 1) \\ \end{eqnarray}$$

Can you find a general rule for $P_n$?

By writing a formula for the nth triangular number $T_n$, show that all pentagonal numbers are one third of a triangular number and prove that the triangular numbers involved are all of the form $T_{3n-1}$