Date: Wed, 14 May 97 17:43:52 CDT
From: rusin (Dave Rusin)
To: pshapiro@oz.sunflower.org
Subject: Re: Constructing a Pentagon
Newsgroups: sci.math

In article <862974232.16174@dejanews.com> you write:
>Does anyone know how to construct a pentagon using a compass and
>straight-edge?	Please e-mail me if you do.

The proof that the regular  n-gon  is constructible by compass
and straightedge only for certain  n  is due to Gauss; presumably
there's a simple construction at least that old. All you need is a
way to construct a 72-degree angle. This will do if you haven't 
received any other ideas:

Start with any line segment  OB. (A few cm in the middle of a sheet of
	notebook paper works out OK). (O will be a vertex of the pentagon).
Drop a perpendicular bisector CAC'  through  OB  so that  AC  and  AC'  both
	have the same length as  OB. (A  is the midpoint of  OB)
Draw the circle  S  at  C  passing thru  O. (This will be the circumscribed
	circle for the pentagon.)

Now get that angle:

Draw enough of the circle at  O  with radius  AO  so you can find the
	intersection  D  with the segment  OC.
Draw a long line  L  perpendicular to  CD  at  D.
Draw enough of the circle at  C  with radius  CC'  so you can find its
	intersections  E, F  with  L.

Both angles  OCE  and OCF  are now  72 degrees. The intersections  E' and F'
	of  S  with  CE and CF respectively give two more vertices of the
	pentagon.
You can get the last two vertices G,H  by intersecting  S  with arcs at
	E' and F'  of length  OE' (=OF'). As a check, the distance GH
	should also equal  OE'.

Of course, it's easier to get a pentagon simply by tying an overhand knot
into a thin strip of paper and then pressing the whole thing flat!

dave