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?
M is any point on the line AB. Squares of side length AM and MB are
constructed and their circumcircles intersect at P (and M). Prove
that the lines AD and BE produced pass through P.
The circumcentres of four triangles are joined to form a
quadrilateral. What do you notice about this quadrilateral as the
dynamic image changes? Can you prove your conjecture?
If the areas are in the ratio 1:2:3 what can you say about the
ratio of the radii?
Can you find some key right-angled triangles that will give you
the lengths you are after? Pythagoras' theorem is useful!
Some useful constructions can be found in the Thesaurus if you
look up "construction". A perpendicular line might be particularly