Example 1 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, Luo Jij Province Mathematics Competition)
Proof: According to the geometric meaning of the condition, using the Pythagorean theorem, starting from a right-angled triangle, construct a right-angled trapezoid \(CDEA\) as shown in the figure. Here, \(CD\) and \(AE\) are the legs, with \(CD \leqslant AE\), i.e., \(m+n \leqslant \sqrt{2} p\). When \(m=n\), the right-angled trapezoid becomes a rectangle, i.e., \(CD=AE\), the equality holds. Therefore, the minimum value of \(\frac{p}{m+n}\) is \(\frac{\sqrt{2}}{2}\).

Analytic geometry is a model of the integration of numbers and shapes. Therefore, when converting some algebraic or trigonometric problems into geometric problems, one should pay attention to flexibly utilizing relevant knowledge of analytic geometry.