In article <firstname.lastname@example.org>, quasi <email@example.com> 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 >two adjacent vertices.
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.