Having trouble solving this Math contest problem.
Prove that there are infinitely many points on the unit circle such that the distance between any two of them is a rational number.
It's obvious to see that there are infinitely many points on a unit circle such that the distance between two points on a unit circle is rational and it's also obvious that there are infinitely many rational distances between two points. But how in the world can you construct an infinite set of points such that the distance between any two points is rational?
I also have another question.
Are there infinitely many rational distances between two points on a unit circle?
I tried answering this one but I am getting two different answers.
On the one hand, WLOG consider one point on $(1,0)$ if not rotate the circle so that this is true. The distance function between any point on the circle and $(1,0)$ is continuous and it produces every real number from $[0,2]$, since rationals are dense in reals there are infinitely many rational distances between two points on a circle.
Secondly, I tried a different approach which provides a different solution:
The distance between any two points $\sin(\alpha), \cos(\alpha)$ and $\sin(\beta),\cos(\beta)$ (where $\alpha, \beta$ are angles made with the positive $x$-axis) is $\sqrt{2(1-\cos(\beta-\alpha))}$ (from the cosine rule). This expression is rational only if the $\cos(\beta-\alpha)$ is rational and the value of the expression is supposed to be a square of a rational. According to Niven's theorem, $\pi/3$ is the only value in which the $\cos(theta)$ is rational. and no value of $\theta$ can make the expression rational.
I don't understand why there is a contradiction.
