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?
It seems at first as though there's not enough information to
fix the angle whose size we are asked to establish - parts of the
construction seem to have a lot of freedom to wander.
But as this situation is explored more it becomes apparent that
the angle of interest maintains its size wherever the unconstrained
parts of the diagram happen to rest.
This freedom to wander may be clear to the group almost
immediately but if it is not, ask the students to reproduce the
figure using the given values. This should help them appreciate
what hasn't been specified, and lead them to question whether it is
necessary for these to be specified.
Now they have something to explore.
Dynamic Geometry may help with enquiry, but isn't essential. Two
drawings of the figure with measurements of the angle should be
enough to suggest a possible general result which can then be
In conjunction with the problem presented the following
connected result can be included : allow one fixed length chord to
move around a given circle, relative to a second chord of some
other fixed length. Joining the end of each chord to the opposite
end of the other will produce two diagonal lines which, it turns
out, intersect at the same angle regardless of the relative
position of the chords.