So here's my problem: Imagine we want to proof some statement about a line, for example it's equation. What a typical textbook would do is to draw a line, then take a point on a line in the I quadrant and then do some constructions and finally prove the statement. Here's where my problem lies: Why do authors only prove the statement for a point that lies in the I quadrant? (or only one point in any other quadrant). So what I mean is this: we have a line that passes through any 3 quadrants but textbooks prove the statements only for one of those quadrants.
So they prove for a point in one of the quadrants and then say "So for any point on the line we have that...".. But why? I mean a point in the I quadrant has different properties that a point in the II, right? So in the first, x>0 and y>0 but in the second x<0.
So why don't they prove for other quadrants? Is there's some neat trick, an argument for not doing that? Down below there's an image from an old textbooks (I tried modern ones, they are much worse) with a proof of equation for a line. I already "proved" it for most of the possibilities using somekind of shortcut, a method that wasn't presented in the textbook, though I am not sure if it's actually valid or not. Anyways, I'd like to know how would you prove the statement(in case if we really need to prove all of them), I mean none of us would like to prove the statement for all the 12 possibilities, right?(also look another image).
Also, most of the time they took only one line, so for example a line that passes trough II, I and IV quadrants and prove a statement for that line, and then say "For any line that passes the x axis at a and y-axis at b we have that...". So, in essence they proved the statement for a line that passes through positive x and y axes. Obviously not any line passes through positive parts of the axes.
So in the last image you can see we have 4 possible lines that pass through both axes(not counting the ones that pass through the origin.). And on each of them we have 3 points that lie in different quadrants.
Thank you in advance.
y = y(1) + y(2); where y(1) comes from applying the sine law to $\triangle BPM$ and y(2) = c.