### 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?

### DOTS Division

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

### Loopy

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

##### Age 14 to 16 Challenge Level:

You may find it useful to print off some dotty paper for this task.

Draw any convex quadrilateral and find the midpoints of the four edges.
Join the midpoints to form a new quadrilateral.

Try it a few times starting with different convex quadrilaterals.

What do you notice about the new quadrilaterals formed by joining the midpoints?
Does this always happen?

Can you find a counter-example?
Can you make a conjecture?
You may wish to use the GeoGebra interactivity to explore.