Example 11. Let $m, n, p$ be positive real numbers, and $m^{2}+n^{2}-p^{2}=0$. Find the minimum value of $\frac{p}{m+n}$. (1988, Sichuan Province Junior High School Competition)
Solve As shown in the figure, construct a square $ABCD$ with side length $m+n$. Points $E, F, G, H$ divide each side of the square into segments of lengths $m$ and $n$. Clearly, $EFGH$ is a square with side length $p$.

From the figure, it is evident that $EG = \sqrt{2} p$, and $EG \geqslant AD$ (equality holds when $DG = AE$, i.e., when $m = n$).
$$
\therefore \sqrt{2} p \geqslant m+n \text {, }
$$
or $\frac{p}{m+n} \geqslant \frac{\sqrt{2}}{2}$ (equality holds when $m = n$).
Thus, the minimum value of $\frac{p}{m+n}$
$$
\text { is } \frac{\sqrt{2}}{2} \text {. }
$$