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?
Samantha and Shummus both realised that
in order to create a triangle with a right angle, the band had to
go through the centre of the circle. Shummus writes:
Xianglong Ni notes that:
If we have 9 points on the circle then you can't create a
right-angle using the points. This is so because a right angle is
inscribed in a semicircle; It is facing a diameter. But you can
only create a diameter when there is an even amount of points on
the circle. If the number of points on the circle is even then yes.
If the number is odd then no.
Rachel from Newstead sent us a few
diagrams to illustrate examples of right-angled triangles in
circles with an even number of points.