Angle trisection and the regular nine-gon
We know that it is impossible to trisect an angle using only a straight-edge
and compass. Since Geometer's Sketchpad mimics such constructions,
one cannot write a script or sketch that trisects an arbitrary angle (using
only the buttons and construction pull down menu in sketchpad.) However,
one can create a sketch that mimics Archimede's trisection using a notched
straight-edge. Such a sketch is below. The one step that is
not a valid construction must be done by hand.
|In the sketch at the right, select the (acute) angle CAB to
be trisected by moving the point C. Now, move P so
that the line PQ goes through C. The angle CQB
is one third of the angle CAB.
The step that must be done by hand -- moving P so that PQ
goes through C -- is the step which is not a valid construction.
The regular pentagon is constructible. Thus, one can write a sketch
which produces a regular pentagon inscribed in a given circle (see below).
The regular nine-gon, on the other hand, cannot be constructed using only
a straight-edge and compass. But, one can use Archimedes' construction.
This is done below. Again, one can adjust the circle in which a regular
nine-gon is to be constructed by moving A and B. Try
doing this. Note how the figure is distorted. Now, adjust P
so that the line PQ goes through C. This gives a regular
This java applet was produced using JavaSketchpad. The following
is the maker's beta on this product:
This is a prototype of JavaSketchpad,
a World-Wide-Web component of The
Geometer's Sketchpad. Copyright ©1990-1998 by Key Curriculum
Press, Inc. All rights reserved. Portions of this work were funded by the
National Science Foundation (awards DMI 9561674 & 9623018).
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This page was created November 26th, 1998.