
Re: rational ngon inscribed in a unit circle
Posted:
Dec 15, 2013 1:59 AM


On 12/14/2013 06:07 PM, Richard Tobin wrote: > In article <qjfpa9l5bg32acti52rrl1cob63ic5e41t@4ax.com>, > quasi <quasi@null.set> wrote: > >> If a pentagon inscribed in a unit circle has rational edge >> lengths, then for some reordering of the edges within the >> circle, some diagonal has rational length. > > What do you make of > > radius 168: 53 91 187 292 294 > >  Richard >
I find that it's a tiny bit off, when adding up the angles. The halfedges are: 53/2, 91/2, 187/2, 292/2 and 294/2.
If theta_j is the j'th halfangle for the j'th isosceles triangles, j = 1 ... 5, then
sin(theta_1) = (53/2)/168 sin(theta_2) = (92/2)/168 sin(theta_3) = (187/2)/168 sin(theta_4) = (292/2)/168 sin(theta_5) = (294/2)/168
Then solving for the theta_j, j=1 ... 5, and addingup the halfangles:
? asin(53/336)+asin(91/336)+asin(187/336)+asin(292/336)+asin(294/336) %6 = 3.14159265358981471905
? Pi %7 = 3.14159265358979323846
The difference with pi is about 2.1e14. The sum of the center angles of the five isosceles triangles should be 360 degrees or 2pi; the sum of the five halfangles should be 180 degrees or pi.
David Bernier
 http://www.bibliotecapleyades.net/sociopolitica/last_circle/1.htm

