Four rods of equal length are hinged at their endpoints to form a
rhombus. The diagonals meet at X. One edge is fixed, the opposite
edge is allowed to move in the plane. Describe the locus of the
point X and prove your assertion.
In a right angled triangular field, three animals are tethered to posts at the midpoint of each side. Each rope is just long enough to allow the animal to reach two adjacent vertices. Only one animal can reach all points in the field. Which one is it and why?
This gives a short summary of the properties and theorems of cyclic quadrilaterals and links to some practical examples to be found elsewhere on the site.
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.