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.