### Is There a Theorem?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

### AP Rectangles

An AP rectangle is one whose area is numerically equal to its perimeter. If you are given the length of a side can you always find an AP rectangle with one side the given length?

### Pick's Theorem

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

# Coins on a Plate

##### Stage: 3 Challenge Level:

Here we use excerpts from several different proofs.

We start with the solution fromMolly and Catherine of Mount School, York .

Points C, A and A' are co-linear. To explain this, consider the common tangent to both circles at the point of contact. It is both perpendicular to CA' and to AA' so, as there is a common point in A' , the initial statement is demonstrated.

Molly and Catherine's method proves, in the same way, that C, B and B' are co-linear and A, D and B are co-linear. They finished the proof correctly as did Sarah and Caroline of Ipswich High School and James of Hethersett School, Norfolk. For the following method you may visualise swinging AD round to AA' and BD round to BB'.

Next is the solution byCharlotte and Frances of Ipswich High School

It can be proved that the triangle ABC has perimeter equal to the diameter of the circle centre C because AD = AA' making CAD = CAA' and BD = BB'

so CBD = CBB' . This proves that the perimeter of the triangle ADBCA equals the diameter of the circle because CAA' and CBB' are both radii.

Clare of Maidstone Girls' Grammar School used much the same method as Charlotte and Frances.