[Example 2.4.8] Given the three sides of a triangle $a, b, c$, let $S$ denote the semi-perimeter, i.e., $S=\frac{1}{2}(a+b+c)$. Construct a triangle with side lengths $S-a, S-b, S-c$, and continue this process until the given side lengths can no longer form a new triangle.
Question: For which initial triangles can this construction process continue indefinitely?
We study a general case, that is, the change in the relationship of the side lengths after constructing the triangle for the $(k+1)$-th time, given that we have already constructed it $k$ times.

Let $\left(a_{k}, b_{k}, c_{k}\right)$ be the side lengths of the triangle after $k$ constructions, and $S_{k}$ be its semi-perimeter, then
$$
\begin{aligned}
a_{k+1} & =S_{k}-a_{k}, \\
b_{k+1} & =S_{k}-b_{k}, \\
c_{k+1} & =S_{k}-c_{k} .
\end{aligned}
$$
Adding (1), (2), and (3) gives
$$
2 S_{k+1}=3 S_{k}-\left(a_{k}+b_{k}+c_{k}\right)=3 S_{k}-2 S_{k}=S_{k},
$$

Thus,
$$
S_{k+1}=\frac{1}{2} S_{k} \text {. }
$$

Subtracting (2) from (1) yields
$$
a_{k+1}-b_{k+1}=b_{k}-a_{k},
$$

which means
$$
\left|a_{k+1}-b_{k+1}\right|=\left|b_{k}-a_{k}\right|,
$$

From this, we derive a general rule: after constructing the $(k+1)$-th triangle from the $k$-th triangle, the semi-perimeter is half of the original semi-perimeter, and the difference in side lengths remains unchanged.
For the initial triangle, we set
$$
\left|a_{0}-b_{0}\right|=|a-b|, S_{0}=S,
$$

Then, after $k$ constructions, according to the above rule, we have
$$
\begin{array}{c}
\left|a_{k}-b_{k}\right|=|a-b|, \\
S_{k}=\frac{1}{2^{k}} S .
\end{array}
$$

Note that $|a-b|$ is a constant.
If $|a-b| \neq 0$, then for sufficiently large $k$, we must have
$$
S_{k}<|a-b|=\left|a_{k}-b_{k}\right|,
$$

which is impossible.
Therefore, we must have $|a-b|=0$, i.e., $a=b$.
Similarly, we must also have $a=c$,
Thus,
$$
a=b=c .
$$

Therefore, the initial triangle is an equilateral triangle.
In solving this problem, we used a general approach to handle the construction of the triangle, and derived the final result using a general conclusion.