In a Cartesian coordinate system, gcd(a,b) is equal to the number of vertices (points with integral coordinates) which lie on the segment with extremities $(0,0)$ and $(a,b)$ (excluding $(0,0)$).
How could I prove it? I have an idea why this works, but it is not clear to me how I could prove it. If you could give me a proof, suggest a book or article where this geometric interpretation appears, I would greatly appreciate it.I've been searching but still can't get anything
The segment has the equation $(x,y)=\lambda(a,b)=\lambda\gcd(a,b)\,\left(\dfrac a{\gcd(a,b)},\dfrac b{\gcd(a,b)}\right)$ with $\lambda\in [0,1]$. So $x,y$ are integer every time $\lambda\gcd(a,b)$ is an integer.