Example 1 Given three distinct rational numbers $a, b, c$ satisfying $|a-b|+|b-c|=|a-c|$. Try to compare the sizes of $a, b, c$.

保留源文本的换行和格式，直接输出翻译结果。
Explanation: We can first place the three numbers to be compared, $a$, $b$, and $c$, on a number line, and then make judgments based on the given conditions.

Since $|a-b|+|b-c|=|a-c|$, the positions of $a$, $b$, and $c$ on the number line can be in two situations (as shown in Figure 1(a) and (b)).
(a)
(b)
Figure 1
By observation, we can see:
(1) $a<b<c$; (2) $c<b<a$.