How Does the Triangle Inequality Transform from Equality to Inequality?

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The discussion focuses on the transformation of the triangle inequality from equality to inequality, particularly examining how taking the absolute value of 2uv affects the inequality. It is noted that before applying the absolute value, 2uv could be negative, which would reduce the expression |u|^2 + 2uv + |v|^2. The conversation also references the Cauchy-Schwartz inequality as a basis for understanding the second inequality. Additionally, the participants discuss why squaring both sides of an inequality is permissible only when both sides are non-negative, using an example to illustrate the potential issues with negative values. Overall, the dialogue emphasizes the mathematical principles underlying the triangle inequality and its implications in metric spaces.
Bashyboy
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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?
 
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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.
 
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?
 
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.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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