Geometric Proof for Slopes (Contined Fractions)

Question

I just started learning about continued fractions, and my lecture had a theorem that estimated the slope $a$ of a given line $L$. This was done in terms of the slope of the point $P$ with coordinates $(q,p)$. The proof in the lecture went through the case when $P$ was below $L$. However what I want to know is what would be a proof if $P$ was above $L$.

Lecture Notes:

Answer 1

The proof of the dual case where $P$ is above $L$ is precisely the same, except you need to reverse the ordering in a few places. In fact,the theorem and proof is almost identical to Theorem 7.2 in Stark's An introduction to number theory. Following the proof Stark remarks

We have of course only proved Theorem 7.2 when P is under L. In this and later theorems where we have the two choices of putting a point over or under a line, we will take one of the choices and leave the other choice for the reader. As expected, the proofs are practically identical with the modifications usually restricted to the words "over" and "above" being interchanged with "under" and "below". There are also occasional changes of directions of inequalities and changes of sign.