Example 2. The vertex angle and base of a triangle - given, when is the area maximum? What is the maximum area?
As shown in Figure 2, for $\triangle BAC$ with a fixed base $BC = a$ and vertex angle $A = a$, the trajectory of vertex $A$ is two circular arcs with $BC$ as the chord and the angle subtended by the arc as $a$. When the moving point travels from $B$ to $C$ along the arc, due to the symmetry, when the moving point reaches the midpoint $A_0$ of $\overparen{BC}$, its distance to $BC$ is the greatest (i.e., the altitude from $A$ to $BC$ is the greatest), hence the maximum area is: $S = \frac{a^2}{4} \operatorname{ctg} \frac{a}{2}$. Therefore, we have:

If the vertex angle and the base of a triangle are fixed, then the area is maximized when the two sides are equal.