On the geometrical interpretation of continued fractions we can see the following facts ($\alpha$ is an irrational number) :
1- The line $y=\alpha x$ never passes through a lattice point $\alpha$ is an irrational number.
2- Considering the convergents of $\alpha$, i.e. the numbers $c_i = \frac{p_i}{q_i}$, the points $(q_{2k-1}, p_{2k-1})$ are the lattice points which all lie below the line $y=\alpha x$; and, the points $(q_{2k}, p_{2k})$ are the lattice points which all lie above the line $y=\alpha x$.
3- a) All the points $(q_{2k-1}, p_{2k-1})$ all lie on a single straight line and all the points $(q_{2k}, p_{2k})$ all lie on a(nother) single straight line; and b) these two mentioned lines approach the line $y=\alpha x$ more and more closely the farther out we go.
All the claims in 1, 2 and 3 are from Felix Klein a popular mathematical expositor. I could prove all but the 3-a.
Please help for a proof of why either of all the relevant lattice points even or odd convergents are on a line? Thank you.
Added; that's enough to prove (for example in case of evens) that $$\dfrac{p_{2k}-p_{2k-2}}{q_{2k}-q_{2k-2}}=\dfrac{p_{2k+2}-p_{2k}}{q_{2k+2}-q_{2k}}$$ but I am stuck in it!