<s>[INST] Proving the equality case in triangle inequality Background
When plotted on a real number line, it may be deduced that if 
$$a,b,c \in \mathbb{R} $$
$$a < b < c$$
then 
$$\left| {a - c} \right| = \left| {a - b} \right| + \left| {b - c} \right|$$
Problem
But the problem is with the proof. How can the above statement be proven true from the properties of order structure and the definition of absolute value? 
 [/INST] Since both sides of the equality are nonnegative, we can square it:
\begin{align}
|a-c| = |a-b| + |b-c| &\iff |a-c|^2 = (|a-b| + |b-c|)^2\\
&\iff a^2-2ac+c^2 = a^2-2ab+b^2+b^2-2bc+c^2 + 2|a-b||b-c|\\
&\iff 0 = b^2 - ab - bc + ac + |a-b||b-c|\\
&\iff 0 = -(a-b)(b-c) + |a-b||b-c|\\
&\iff  (a-b)(b-c) = |a-b||b-c|\\
&\iff a-b \,\text{ and } \,b-c \text{ have the same sign}\\
&\iff a \ge b \ge c \,\text{ or }\,a \le b \le c 
\end{align}
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