## § Triangle inequality

We can write this as:
   *A
b/ |
C*  |
a\ | c
*B

The classical version one learns in school:
$c \leq a + b$
The lower bound version:
$|a - b| \leq c$
This is intuitive because the large value for $a - b$ is attained when $b = 0$. (since lengths are non-negative, we have $b \geq 0$. if $b = 0$, then the point $A = C$ and thus $a = CB = AB = c$.
A/C (b=0)
|
| a=c
|
B

Otherwise, $b$ will have some length that will cover $a$ (at worst), or cancel $a$ (at best). The two cases are something like:
 A
||b
||
c|*C
||a
||
||
B

In this case, it's clear that $a - b < c$ (since $a < c$) and $a + b = c$. In the other case, we will have:
 C
b||
||
A|
||
||a
c||
||
||
||
B

Where we get $a - b = c$, and $c < a + b$. These are the extremes when the triangle has zero thickness. In general, because the points are spread out, when we project everything on the $AB=c$ line, we will get less-than( <=) instead of equals ( =).