Iterated Circumcentres - Page 2
The Iterated Circumcentres Rule:
For most positions of P0, P1, P2,
there is a unique circle which goes through these three points.
The centre of this circle is called the
circumcentre of P0, P1, P2.
Point P3 is the circumcentre of P0, P1, P2.
Point P4 is the circumcentre of P1, P2, P3.
In general P(i) is the circumcentre of P(i-1), P(i-2), P(i-3),
for i=3,4,5,...
Thus the resulting infinite sequence of points is entirely determined by the
positions of of P0, P1, P2.
One day I asked myself: What happens?
A good first step is to experiment.
You might try sketching the sequence on paper.
It is difficult to do this accurately.
Instead, it is better to use a lovely Geometry Applet
written by
David E. Joyce, Clark University, Worcester, MA 01610, USA,
which is found at
http://aleph0.clarku.edu/~djoyce/java/Geometry/Geometry.html.
A users manual can be found there.
Try dragging around the three red points.
Do you see any patterns?
The points sometimes appear to converge closer and closer together,
and sometimes they diverge right off the screen.
Do you notice some points lining up?
Why does one of the points seem to not follow the pattern?
(The answer to this last question can be found off of page 4.)
If you would like to plot more points on a bigger screen,
with a protractor to help measure angles,
then check out the
Experiment Room.
When you are ready for another mystery,
go on to Page 3.
or back to Page 1.