Example 1. Given a cyclic quadrilateral $A B C D$ with side lengths $a, b, c, d$, and the longest side length is $a$. Under what conditions on $a$, $b, c, d$ does the perimeter of the cyclic quadrilateral $A B C D$ have a minimum value? And find this minimum value.
Let the area of quadrilateral $ABCD$ be $S$, the radius of the circumcircle be $R$, and $AB=a, BC=b, CD=c, DA=d$ (as shown in the figure). Then,
$$
\begin{aligned}
S & =\sqrt{p(p-a)(p-b)(p-c)} \\
(p= & \left.\frac{1}{2}(a+b+c)\right),
\end{aligned}
$$
$$
\begin{array}{l}
\sin A=\frac{2 S}{a d+b c}, \sin B=\frac{2 S}{a b+c d}, \\
R=\frac{1}{4 S} \sqrt{(a b+c d)(a c+b d)(a d+b c)} .
\end{array}
$$

Therefore, $AC=2 R \sin B=\sqrt{\frac{(a c+b d)(a d+b c)}{a b+c d}}$.
Since the maximum side length of the cyclic quadrilateral $ABCD$ is $a$, the center of the circle is inside the quadrilateral $ABCD$ $\Leftrightarrow \angle BCA$
$$
\begin{array}{l}
<90^{\circ} \Leftrightarrow AB^{2}<BC^{2}+AC^{2} \Leftrightarrow a^{3}<c\left(b^{2}+c^{2}\right. \\
\left.+a^{2}\right)+2 b c d .
\end{array}
$$

From the corollary, we know that the perimeter of the cyclic quadrilateral $ABCD$ has a minimum value if and only if $a^{3}<a\left(b^{2}+c^{2}\right.$ $\left.+d^{2}\right)+2 c d$. The minimum value is
$$
2 AC \sin A=4 S \sqrt{\frac{a c+b d}{(a b+c d)(a d+b c)}} .
$$