About the vertices of a regular polygon in the plane having rational coordinates

Question

I have to prove that, except in the case $n=4$, the vertices of a regular $n$-agon in the Euclidean plane cannot have all rational coordinates $(x,y)$.

Some idea?

Answer 1

Assuming that all the vertices of a regular $n$-agon ($n\neq 4$) have rational coordinates, by the shoelace formula the area of such a polygon is a rational number. On the other hand, the area is given by:

$$ A=\frac{nl^2}{4}\,\cot\frac{\pi}{n} $$ but $\cot\frac{\pi}{n}$ is an irrational number for every $n\geq 3,n\neq 4$: that leads to a contradiction.