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