Is the following approach mathematically correct? (While solving questions)

Question

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?

Answer 1

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.

Answer 2

"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.