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Re: Req: Pentagon problem  help please !
Posted:
Aug 6, 1998 10:21 AM


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   \ WYOXE         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 edgelength 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 angletrisector which you can find "The Book of Numbers" that I wrote with Richard Guy. That book also gives an angletrisector 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 11gon there's a construction using an anglequinquesector, but it was too complicated for us to put into the book).
John Conway



