Example 1. In $\triangle A B C$, $O$ is the circumcenter of $\triangle A B C$, $I$ is the incenter, and $\angle B O C=$ $\angle B I C$. Find $\angle A$.

保留源文本的换行和格式，直接输出翻译结果。
Since $I$ is the incenter of $\triangle A B C$,
$$
\begin{array}{l}
\therefore \angle I B C+\angle I C B \\
=\frac{1}{2}(\angle B+\angle C) \\
=\frac{1}{2}\left(180^{\circ}-\angle A\right),
\end{array}
$$

Therefore, $\angle B I C=180^{\circ}-(\angle I B C+\angle I C B)$
$$
\begin{array}{l}
=180^{\circ}-\frac{1}{2}(180-\angle A) \\
=90^{\circ}+\frac{1}{2} \angle A .
\end{array}
$$

Since $O$ is the circumcenter of $\triangle A B C$,
$$
\begin{array}{l}
\therefore \angle B O C=2 \angle A . \\
\because \angle B O C=\angle B I C, \\
\therefore 90^{\circ}+\frac{1}{2} \angle A=2 \angle A .
\end{array}
$$

Therefore, $\angle A=60^{\circ}$.