Distance between two points in the plane

Question

my teacher asked in the class today the following question: There exists an infinite set M of points in the plane with the property that any three points are non-collinear and such that the distance between any two points is a rational number? Any ideas?

Answer 1

There is such a set $M$. For example, let $M$ be the set of points $(\cos\theta,\sin\theta)$ such that $\tan(\theta/4)$ is rational. Since $\cos x$ and $\sin x$ are rational functions of $\tan(x/2)$, it follows that for any two points $(\cos\theta,\sin\theta)$ and $(\cos \phi,\sin\phi)$ in $M$, we have $\cos(\theta/2)$, $\sin(\theta/2)$, $\cos(\phi/2)$, $\sin(\phi/2)$ all rational.

The distance between the two points is $2|\sin(\theta-\phi)/2)|$. By the expression for the sine of a sum, this is rational.

Remark: The above argument can be found, among other places, in Wikipedia.

It can be shown that if we have infinitely many points, with all distances integers, then the points must lie on a line.