What do you mean "by coordinates"? If you are asked to show that three points are on a line, having an accurate diagram is not enough, is it? Isn't the task to use the constraints presented (these lines are parallel, this is a right angle, etc) in a step-based reasoning for whatever you're asked to prove?

It is done using symbolic algebra, not just assigning specific coordinates if that's what you're asking.

Let's say you're given three parallel lines. You can put your x axis along the first line. Then its equation is y = 0. The other two lines necessarily have equations y = a and y = b for some reals a and b. Then you calculate the other quantities involved via a and b and other parameters you have to introduce. At the end you calculate the coordinates of your three points and verify, symbolically, that they lie on the same line.

Yes, that's right. You end up with a system of equations and solve it. So what does "by coordinates" mean? Is the original commenter simply saying that the machine will always be able to solve that system? He's right if that is what he means. Originally it sounded like "the machine can read the diagram accurately".

>Is the original commenter simply saying that the machine will always be able to solve that system?

Yes.

Well, you can do that reasoning algebraically as equations on the coordinates. So you translate the constraints into equations. See https://en.wikipedia.org/wiki/Analytic_geometry

If memory serves right, there's a certain subset of geometry that decidable in that way. Ie you can just run an algorithm.