Triangle inequality, parallelogram equality

[asdf1]

what does a triangle have to do with triangle inequality, and what does a paralllelogram have to do with parallelogram equality?
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[EnumaElish]

Triangle ineq.

Parallelogram ineq.

 

[asdf1]

i'm still confused:
"triangle inequality is the theorem stating that for any triangle, the measure of a given side must be less than the sum of the other two sides but greater than the difference between the two sides."
yet "|x+y| ≤ |x|+|y| " shouldn't mean "|z|≤ |x|+|y| "?

 

[d_leet]

i'm still confused:
"triangle inequality is the theorem stating that for any triangle, the measure of a given side must be less than the sum of the other two sides but greater than the difference between the two sides."
yet "|x+y| ≤ |x|+|y| " shouldn't mean "|z|≤ |x|+|y| "?

If you think of x and y as vectors in space they will form a triangle with a third vector that is the vector sum x+y so the inequality makes sense.

 

[asdf1]

thank you very much!