I have some misunderstanding with the process of proving statements in coordinate geometry.
Here what I mean: In textbooks when it comes to proving something, the general process of proving (most of the time) happens in only one quadrant. Authors prove some statements using properties of exactly that quadrant(say the first one, where for any (x,y), x > 0, y > 0.). For some reason they do not consider the other quadrants, while saying that these(almost any) statements are true for any two(or three, doesn't matter) points in the coordinate system, or for any line, for any line segment, etc.
As an example, consider the proof of an equation of some line. Authors might take such point (x,y) that x > 0 and y > 0 and then prove the equation (as they say "for every point"). But doesn't the points (x,y) where x < 0 and y > 0 would to have different properties than the one in the First quadrant?
To me it looks like if the author "proved" some statement about all parallelograms by also using the properties of rhombuses in his proof.
Here is my question:
Is there is a reason, why shouldn't I bother about proving some statements for points lying in different quadrants? If there is no such reason, I have to manually prove statements for different quadrants or perhaps there's some neat shortcut?
Thank you in advance.