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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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 Richard Tobin Posts: 1,436 Registered: 12/6/04
Re: rational n-gon inscribed in a unit circle
Posted: Dec 14, 2013 2:27 PM

In article <9j5pa9dtoo4f5af5iko7i3lh0gob3qq4km@4ax.com>,
quasi <quasi@null.set> wrote:

>>Given a set of rational sides, how in general do you determine
>>whether any of the diagonals are rational?

>Since the radius and the edge lengths are known, the sine and
>cosine of each of the central angles to the edges can be found.
>If two edges are placed adjacent, trig sum formulas can be used
>to find sine and cosine of the combined angle. The law of
>cosines can then be used to get the length of the chord between

Right, if the radius is r and the sides are a and b, the diagonal is

a sqrt(1 - b^2 / 4r^2) + b sqrt(1 - a^2 / 4r^2)

But how do you determine whether that is rational?

Ah, I see that it's a well-known fact that the sum of the square roots
of x and y is rational iff x and y are both squares of rational
numbers.

So the diagonal is irrational if either 4r^2 - b^2 or 4r^2 - a^2
is not a perfect square.

-- Richard

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