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:
I noticed that the bands had to be started in the centre.
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.
Indika of Helena Romanes 6th Form
College sent us her explanation for why the band must go through
the centre of the circle:
The only way that a right angle triangle
can be created between 3 points round the edge is when the angle
subtended at the centre by two of the points is 180 degrees, this
therefore proves that two of the points have to be opposite each
other (this means having an equal number of pegs).
If you haven't met this idea before, you
may want to look at another problem from August
Here are another couple of examples of
right-angled triangles using the same eight-point and ten-point
circles that Rachel used:
This is because the angle subtended at the centre by two points are
exactly double the angle subtended at the edge by the same points.
This rule will apply to all circles, i.e. there will be a right
angled triangle if two pegs are placed opposite each other.