### 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 shows the groupings of the numbers which is mirrored in the derivation of the formulae.
The rule is generalisable but can you convince us why?

For the last part you need a formula for triangular numbers. Each triangular number is the sum of all the whole numbers so the fifth triangular number is 1+2+3+4+5.

By reversing and adding any group of consecutive numbers to itself you can generate the triangular numbers. The thesuarus might help here.

You might find it helpful to visualise the pentagonal numbers as made from triangular numbers.

Now you should notice that the formula for the pentagon numbers can be written in terms of triangular numbers.

There is a little bit of agebraic substitution necessary to get you to the point where you can show that every pentagonal number is a third of a triangular number.

Is the inverse the case?