Example 1
If $\odot(O, r)$ can be embedded in $\triangle A B C$, what is the maximum value of $r$?
Intuitively, among the circles that can be inscribed in $\triangle A B C$, the radius of the incircle of $\triangle A B C$ is the largest. This fact can be proven as follows:

Let $\odot(O, r)$ be inscribed in $\triangle A B C$ (Figure 2.1). Then the distances from $O$ to the sides of $\triangle A B C$ are all greater than or equal to $r$, so
$$
\begin{aligned}
\triangle A B C & =\triangle O B C+\triangle O A C+\triangle O A B \\
& \geqslant \frac{1}{2} r a+\frac{1}{2} r b+\frac{1}{2} r c \\
& =\frac{1}{2} r(a+b+c),
\end{aligned}
$$

thus
$$
r \leqslant \frac{2 \triangle A B C}{a+b+c} .
$$

The equality holds if and only if $\odot(O, r)$ is the incircle.