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?

Coins on a Plate

Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.

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?

Triangles' Triangle

Stage: 3 Short Challenge Level:

The perimeter of the large triangle is $24 \; \text{cm}$, so it has side of length $8 \; \text{cm}$, so each small triangle has side of length $1 \; \text{cm}$. So each small triangle has perimeter $3 \; \text{cm}$

\begin{eqnarray}\text{Total length of black lines} &=& \text{Total perimeter of shaded triangles} \\ &=& \text{Number of shaded triangles} \times 3\; \text{cm} \\ &=& 27 \times 3\; \text{cm}\\ &=& 81\; \text{cm} \end{eqnarray}

This problem is taken from the UKMT Mathematical Challenges.