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?
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.
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?
Tom and Jerry started with identical rectangular sheets of paper. Each of them cut his sheet into two. Tom obtained two rectangles, each with a perimeter of $40$cm while Jerry obtained two rectangles, each with a perimeter of $50$cm. What was the perimeter of Tom's original sheet of paper?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
This problem is taken from the UKMT Mathematical Challenges.