§ Triangle inequality

We can write this as:
   *A
 b/ |
C*  | 
 a\ | c
   *B
The classical version one learns in school:
ca+b c \leq a + b
The lower bound version:
abc |a - b| \leq c
This is intuitive because the large value for aba - b is attained when b=0b = 0. (since lengths are non-negative, we have b0b \geq 0. if b=0b = 0, then the point A=CA = C and thus a=CB=AB=ca = CB = AB = c.
A/C (b=0)
|
| a=c
|
B
Otherwise, bb will have some length that will cover aa (at worst), or cancel aa (at best). The two cases are something like:
 A
 ||b
 ||
c|*C
 ||a
 ||
 ||
 B
In this case, it's clear that ab<ca - b < c (since a<ca < c) and a+b=ca + b = c. In the other case, we will have:
 C
b||
 ||
 A|
 ||
 ||a
c||
 ||
 ||
 ||
 B
Where we get ab=ca - b = c, and c<a+bc < a + b. These are the extremes when the triangle has zero thickness. In general, because the points are spread out, when we project everything on the AB=cAB=c line, we will get less-than(<=) instead of equals (=).