Geometric proof for properties of Farey sequence

Question

Let $P=(a,c)$ and $P^{'}=(b,d)$ be integral co-ordinates such that $\frac{c}{a}$ and $\frac{d}{b}$ are consecutive terms of Farey sequence. If $O$ is the origin how do I prove no integral co-ordinate can lie inside the triangle $OPP^{'}$ ?

Answer 1

A lattice point is a point with integer coordinates. You already know that $\frac{c}{a}$ and $\frac{d}{b}$ are in simplest form. Thus, no lattice points lie on the segments $\overline{OP}$ or $\overline{OP'}$. Assume that we have proven that no lattice points lie on the segment $\overline{PP'}$. (I assume you have already done this based on your recent edit to your question.)

By Pick's theorem, the area of a triangle with lattice point vertices is $A=i+\frac{b}{2}-1$, where $i$ is the number of interior lattice points and $b$ is the number of lattice points on the boundary. From the previous paragraph, we know that $b=3$. If we can show that the area of $\triangle OPP'$ is $\frac12$, then we have shown that $i=0$, which is what we want to prove.

The area of $\triangle OPP'$ is $\frac12(ad-bc)$. (See here.) But since $\frac{c}{a}$ and $\frac{d}{b}$ are Farey neighbors, we have $ad-bc=1$.