### 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 Triangles

##### Stage: 4 Challenge Level:

This diagram shows how the first triangular number can be added to 3 copies of the second triangular number to make the fourth triangular number:

That is: $$T_1 + 3 \times T_2 = T_4$$ Here is a diagram showing how the second and third triangular numbers can be combined to make the sixth triangular number:

$$T_2 + 3 \times T_3 = T_6$$ Can you generalise this rule?

Can you find a rule in terms of $T_n$ and $T_{n+1}$?