Example 3. If $a, b, c$ are non-negative real numbers, find
$$
\begin{array}{l}
\sqrt{a^{2}+b^{2}} \\
+\sqrt{b^{2}+c^{2}} \\
+\sqrt{c^{2}+a^{2}} \text { 's minimum }
\end{array}
$$

value.
Let point $\mathrm{A}$
$(a, b)$, point $B(a+$
$b, b+c)$, point $C(a+b+c, a+b+c)$, then
$$
\begin{array}{l}
|O A|=\sqrt{a^{2}+b^{2}},|A B| \\
=\sqrt{b^{2}+c^{2}},|B C|=\sqrt{c^{2}+a^{2}}, \text { and } \\
|O C|=\sqrt{2}(a+b+c) .
\end{array}
$$
$\because$ Connecting $\geqslant 1$ OC:
$$
\begin{array}{l} 
\therefore \sqrt{a^{2}+b^{2}}+\sqrt{b^{2}+c^{2}}+\sqrt{c^{2}+a^{2}} \\
\geqslant \sqrt{2}(a+b+c), \text { so } \sqrt{a^{2}+b^{2}} \\
+\sqrt{b^{2}+c^{2}}+\sqrt{c^{2}+a^{2}} \text { has a minimum value of } \\
\sqrt{2}(a+b+c) .
\end{array}
$$