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For example, consider this question (I know the question below is nonsense. But it isn't important here, so I will go on) :

' For any $\triangle ABC$, it is known that when a point $G$ inside $\triangle ABC$ is set to satisfy $ \overline {AB} : \overline {GA} = \overline {BC} : \overline {GC} = 3:1$, the ratio $\overline{GB} : \overline {CA}$ is constant. Then, what is the ratio? '

Then I think 'Any $\triangle ABC$ is said to satisfy the given condition. Then, whether I set $\triangle ABC$ as an equilateral one or not doesn't affect anything. So I will set $\triangle ABC$ as an equilateral triangle and go on. '

I have made many correct answers with this approach in quite a lot of tests, but it this approach mathematically correct? I think my solve doesn't lose generality because the question already states the condition is satisfied for general triangles.

But if(and maybe) this approach is not correct, why?

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  • $\begingroup$ What does mean that a question is not true? Statements can be true or false, but not questions. $\endgroup$ – Hume2 May 26 '19 at 12:55
  • $\begingroup$ The approach is correct only on the assumption that the problem is stated correctly, but it may not be. So I would say that it's a good approach to figure out what the ratio is, but for a complete solution, you still need to for a general triangle. $\endgroup$ – saulspatz May 26 '19 at 12:55
  • $\begingroup$ @Hume2 I meant that the 'x:x = x:x = 3:1' part may be not true, since I created the question without considering whether it would be solvable. I deleted the phrase in case it is confusing $\endgroup$ – Verthele May 26 '19 at 13:00
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If it specifically tells you that the ratio is constant, then you're allowed to do this. But otherwise, you can't and you'd need to prove it for every triangle.

In this case, however, you can assume that the triangle is equilateral by taking a suitable affine transformation and noting that ratios of lengths are unaffected. $G$ in the problem happens to be the centroid.

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  • $\begingroup$ Then if the question tells me the ratio is constant, setting ABC as an equilateral one can be a complete solution in itself? $\endgroup$ – Verthele May 26 '19 at 13:03
  • $\begingroup$ @NumberTWO Yes, it’s fine to prove the problem only for equilateral triangles if the problem says the ratio is constant. $\endgroup$ – auscrypt May 27 '19 at 5:59
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"For any" means that all triangles have this feature, so you need to proove it for all triangles. If you proove it only for one triangles, it doesn't proove the whole statement. However, if you need to proove that the statement is false, you can do enough by finding a single triangle for which it is false.

If it said "there exists a triangle such as...", you could do enough by finding a single triangle with that feature. And if you wanted to proove that it's false, you would need to proove it for all triangles.

In your case, you are speaking only about ratios. You can take a linear transformation which transforms the given triangle to an equidistant triangle. Linear transformations preserve ratios, so you could do that but you should clearly specify what you are doing.

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  • $\begingroup$ Then what about cases like this? --- Setting the given triangle as a equilateral one is correct or not? $\endgroup$ – Verthele May 26 '19 at 13:12
  • $\begingroup$ This one is also only about ratios, so yes. $\endgroup$ – Hume2 May 26 '19 at 13:16

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