Why do this problem?
First, the surprising amount of variation in possibilities shown in
the video is worth the journey. Secondly, even though there is such
variation the outer quadrilateral is always cyclic.Finally,
specialising by trying numbers can help form a map of the journey
you need to make in order to prove the generalisation for any
The first stage is simply to investigate:
Construct an image with the given constraints either using dynamic
geometry or with a ruler and compasses. Using ruler and compasses
is difficult simply because you need some flexibility to ensure a
reasonable overlap of the circles.
Are learners surprised by the flexibility visible in the dynamic
Allow time for lots of discussion about construction techniques,
the order of working (formed by the constraints) and the freedoms
available (how many circles will meet the cirteria?).
Now for the problem.
A first step is to encourage exploration by writing in some angle
sizes (following a discussion of the properties of opposite angles
of a cyclic quadrilateral). Does the outside quadrilateral have
opposite angles whose sum is 180 degrees and is therefore
In specialising by using numbers for angles and keeping track of
which angles can be calculated from others, the steps to a
generalisation are much clearer.
- What defines a cyclic quadrilateral?
- What are the freedoms?
- What the contraints?
. This problem may be slow to load.
Focus on the construction and looking at specific examples.
For work on cyclic quadrilaterals try Pegboard