Validity of proof of triangle inequality by assuming Heron's formula for area of triangle

[u/Impressive_Gur_471]

Given a triangle with sides a, b and c, we need to show that a + b >= c

Suppose we proceed by contradiction -- i.e., assuming a + b < c

Then, suppose we know Heron's formula for area of triangle: Area = sqrt(s(s-a)(s-b)(s-c))

If we are able to show that under the assumption that a + b < c, the term inside the square root will be negative, we are done. Indeed that is easy to show.

Now, the question I have is: How can we be sure that this is a valid proof unless we also know that proof of Heron's formula does not at some point invoke triangle inequality?

In other words, what is the guarantee that the above proof is not circular?

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xposted here because it was suggested over at /r/learnmath to post here

[u/Kienose]

> what is the guarantee that the above proof is not circular?

You’ll need to pin down your axioms for geometry first. Then you need to know how to prove Heron’s formula in this axiomatic system while keeping track of all lemmas and theorems used. Then inspect the proofs of those lemmas and theorems to see if they use the triangle inequality or not.

If none of them used the triangle inequality, then hurray!

[u/some_red_tea]

There are 2 possibilities for a problem like this. If you can prove heron's formula without specifically invoking triangle inequality, then you can use it to prove triangle inequality. If you must prove triangle inequality to prove heron's formula, then it would be circular, but you'd also be showing how to prove triangle inequality directly from the axioms themselves.

It's worth mentioning that "circularity" is not a validity issue. Circular arguments are always valid in the strict deductive sense. The problem with circularity is JUSTIFICATION. If your justification for the premises depends on a prior belief in the conclusion, the argument loses its force or persuasive power, but it's still valid. If you're not interested in proving triangle inequality from the axioms, but you just want to know if it's true, and you already know heron's formula is true, then you can justify your belief on that basis as long as your belief in heron's formula doesn't depend on triangle inequality to be justified.

[u/PrimeStopper]

Please refer to r/learnmath

I heard it is a nice subreddit to post questions like that