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. Sorry, this page requires a Java-compatible web browser.

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 nine-gon.
 
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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).

Arthur's Home Page
This page was created November 26th, 1998.
URL:  http://www.nevada.edu/~baragar/geom/nine.htm