Question

What is the difference between a coordinate proof and a proof method?

Answer 1

The difference between the two is that in coordinate geometry, you have some way of finding the coordinates of any part of the shape in question. This is useful when your proof uses the fact that a line (especially one coming from the origin to some coordinate) can be split up into two parts: x and y part. For an example, see the proof of the law of cosines, which uses coordinate geometry in order to prove a more general version of the Pythagorean theorem using trigonometry and the Pythagorean theorem.