Example 1 As shown in Figure 13-4, in $\triangle A B C$, $O$ is the circumcenter of $\triangle A B C$, and $I$ is its incenter. If $\angle B O C=$ $\angle B I C$, find $\angle A$.

保留了原文的格式和换行。
Since $I$ is the incenter, then $\angle I B C + \angle I C B = \frac{1}{2}(\angle B + \angle C) = 90^{\circ} - \frac{1}{2} \angle A$, hence $\angle B I C = 90^{\circ} + \frac{1}{2} \angle A$. Also, $O$ is the circumcenter, so $\angle B O C = 2 \angle A$, thus $90^{\circ} + \frac{1}{2} \angle A = 2 \angle A$, which means $\angle A = 60^{\circ}$ is the solution.