Analytic Geometry proof. Triangle's area in diff. quadrants

[Guest]

Hi, Most of the time authors in textbooks prove statements in analytic geometry like that: they pick up points in the first quadrant(x>0,y>0), and then prove some relationship involving these points. But they almost never pick up points from other quadrants(even if a line or any other structure passes through other quadrants.) I am not satisfied by that situation. And I also do not want to manually prove the formula for point in different quadrants.(You see, there's lot's of combinations, like points may reside in 1,2,3 quadrants, or in 2,3,4, or in 1,2,2, or 3,2,3) So you see how many combinations are out there... So, if there's a shortcut, please, could you explain to me the process of proving the formula for any quadrant? Thanks a lot.

[HallsofIvy]

I have to say that I don't think I am understanding what you are saying- or if I am then what you are saying is not true! The coordinate axes, and the "quadrants" resulting from them are completely arbitrary and the truth of geometric statements is completely independent of the quadrant. Textbooks don't "prove statements using points in the first quadrant", they use arbitrary points, then, perhaps, choose axes to put those points in the first quadrant for convenientce. The truth of geometric statements has nothing to do with what quadrant points are in.