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Topic: rational n-gon inscribed in a unit circle
Replies: 59   Last Post: Dec 19, 2013 12:56 AM

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 David Bernier Posts: 3,892 Registered: 12/13/04
Re: rational n-gon 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 half-edges are: 53/2, 91/2, 187/2, 292/2 and 294/2.

If theta_j is the j'th half-angle 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 adding-up the half-angles:

? 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 2.1e-14. The sum of the center angles
of the five isosceles triangles should be 360 degrees or 2pi; the
sum of the five half-angles should be 180 degrees or pi.

David Bernier

--

Date Subject Author
12/10/13 quasi
12/10/13 ross.finlayson@gmail.com
12/10/13 quasi
12/11/13 quasi
12/11/13 quasi
12/11/13 quasi
12/12/13 quasi
12/12/13 Helmut Richter
12/12/13 quasi
12/11/13 scattered
12/11/13 quasi
12/11/13 fom
12/11/13 fom
12/11/13 quasi
12/11/13 fom
12/11/13 Richard Tobin
12/11/13 Richard Tobin
12/12/13 quasi
12/12/13 Richard Tobin
12/12/13 Richard Tobin
12/12/13 quasi
12/12/13 Brian Q. Hutchings
12/13/13 quasi
12/13/13 Brian Q. Hutchings
12/12/13 Thomas Nordhaus
12/12/13 Richard Tobin
12/12/13 quasi
12/12/13 Richard Tobin
12/12/13 quasi
12/12/13 quasi
12/13/13 Richard Tobin
12/13/13 Richard Tobin
12/13/13 quasi
12/13/13 Richard Tobin
12/13/13 quasi
12/13/13 Richard Tobin
12/13/13 quasi
12/13/13 quasi
12/12/13 Richard Tobin
12/13/13 quasi
12/13/13 Richard Tobin
12/13/13 quasi
12/14/13 quasi
12/14/13 quasi
12/14/13 quasi
12/14/13 quasi
12/14/13 Richard Tobin
12/15/13 quasi
12/15/13 quasi
12/15/13 Richard Tobin
12/15/13 David Bernier
12/15/13 quasi
12/18/13 Richard Tobin
12/18/13 ross.finlayson@gmail.com
12/19/13 quasi
12/14/13 Richard Tobin
12/14/13 quasi
12/14/13 Richard Tobin
12/14/13 ross.finlayson@gmail.com
12/15/13 Brian Q. Hutchings