Example 2. In acute triangle $\triangle A B C$, the distance from vertex $A$ to the circumcenter $O$ is equal to the distance to the orthocenter $H$. Find the possible value of $\angle A$.

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$$
\begin{array}{l}
AI I=R \sqrt{2+2 \cos (\gamma-\beta)}=AO=R, \\
\cos (\gamma-\beta)=-\frac{1}{2} \text { and } 0^{\circ}<\gamma-\beta<180^{\circ}, \\
2 \angle A=\gamma-\beta=120^{\circ}, \angle A=60^{\circ} .
\end{array}
$$