Triangle Inequality Proof

[Bashyboy]

Hello all,

I am currently reading about the triangle inequality, from this article
http://people.sju.edu/~pklingsb/cs.triang.pdf

I am curious, how does the equality transform into an inequality? Does it take on this change because one takes the absolute value of 2uv? Because before the absolute value, 2uv could be a negative value, thus making all of |u|^2 + 2uv + |v|^2 smaller, is this correct?

 

[UltrafastPED]

You are correct ... that is why the first inequality appears. The second one is from the Cauchy-Schwartz inequality, as noted.

These are properties that are required for a metric space.

 

[Bashyboy]

I have one other question. In the article, it says that since both sides of the inequality of non-negative, it is permissible to then square both sides of the inequality. Why would it not be possible to square both sides if both sides were negative?

 

[UltrafastPED]

I'm sure that they said "you can square each term since they are all positive". Try that with this inequality:

1 - 2 < 1 ... hence the requirement for all positive.

 

[CellCoree]

Hi there,

The triangle inequality is a mathematical concept that states that the sum of any two sides of a triangle must be greater than the third side. This can be represented as a mathematical equation: |a| + |b| ≥ |a + b|, where a and b are the lengths of two sides of a triangle and |a| and |b| represent the absolute values of those sides.

In terms of the proof, the transformation from equality to inequality occurs when we take the absolute value of 2uv. This is because, as you mentioned, 2uv could be a negative value and taking the absolute value ensures that it is always positive. This is important in the context of the triangle inequality because we are dealing with lengths, which cannot be negative.

So, to answer your question, the transformation from equality to inequality occurs because we need to ensure that the lengths of the sides are always positive, which is necessary for the triangle inequality to hold true. I hope this helps clarify things for you. Let me know if you have any other questions. Happy learning!