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Many IMO geometry problems involves proving things like: these three points are on one line these three lines intersect at one point these two lines are para

Many IMO geometry problems involves proving things like: these three points are on one line these three lines intersect at one point these two lines are parallel these four points are on a circle these two angles are equal etc ... There are things that absolutely can be done via coordinates, if one can brute force large algebraic expressions.

lordnacho 7y ago
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?
- eru 7y ago
  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 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.
- owaty 7y ago
  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.
  - lordnacho 7y ago
    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".
    - OscarCunningham 7y ago
      >Is the original commenter simply saying that the machine will always be able to solve that system? Yes.

lordnacho 7y ago

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?

eru 7y ago
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 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.
owaty 7y ago
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.
- lordnacho 7y ago
  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".
  - OscarCunningham 7y ago
    >Is the original commenter simply saying that the machine will always be able to solve that system? Yes.