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Topic: Req: Pentagon problem - help please !
Replies: 8   Last Post: Mar 12, 2004 7:15 PM

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John Conway

Posts: 2,238
Registered: 12/3/04
Re: Req: Pentagon problem - help please !
Posted: Aug 6, 1998 10:21 AM
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On 4 Aug 1998, DWill28107 wrote:

> Trying to construct a regular pentagon using only a pair of compasses and a
> straight edge. Can it be done ? My level of math knowledge is high school.



Yes - Let N,S,E,W be the points of a circle with center O in the four
compass directions, M be the midpoint of OE and MX (with X on OE)
the bisector of the angle OME:

N
| |
| M |
| |\|
W-Y----O-X----E
| | |
| | |
| |
S

Then the line through X perpendicular to OE hits the circle
in two points of the regular pentagon that has a vertex at E. You
can either get the other two points by stepping around the circle with
a compass set to the edge-length so found for the pentagon, or by
replacing X in the above by the point Y where the EXTERNAL bisector
of OME meets OW.

> Please e mail me with the solution & proof if possible; or give me an URL where
> I can find the answer on the web.
> And how about septagons....


Well, for one thing the proper name is "heptagon", not "septagon. There
isn't a construction for a regular heptagon using ruler and compass according
to Euclid's rules, but there is a construction using an angle-trisector
which you can find "The Book of Numbers" that I wrote with Richard Guy.
That book also gives an angle-trisector construction that uses ruler and
compasses in a manner not sanctioned by Euclid, so you can combine them to
give such a construction for the regular heptagon. The book also gives
similar constructions for the regular polygons with 13 and 17 sides (for the
regular 11-gon there's a construction using an angle-quinquesector, but it
was too complicated for us to put into the book).

John Conway





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