What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
Two semicircle sit on the diameter of a semicircle centre O of
twice their radius. Lines through O divide the perimeter into two
parts. What can you say about the lengths of these two parts?
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?
The blue square is fixed and the centres of the blue and
yellow squares coincide. The yellow square rotates about its
In the middle ages stone masons used a ruler and compasses
method to construct exact octagons in a given square window.
Open your compasses to a radius of half the diagonal of the
square and construct an arc with centre one vertex of the
square - mark the 2 points where the arc crosses the sides.
Do that for all 4 vertices of the square giving 8 points.
Join these points to draw an octagon. Can you prove that the
octagon is regular?
You might find the proof by relating the construction to the
rotating square. Where does the regular octagon appear when
you rotate the yellow square in this diagram?