Example 2. Extend the altitudes of the acute triangle $\Delta \mathrm{A}_{1} \mathrm{~B}_{1} \mathrm{C}_{1}$ to intersect the circumcircle at $\mathrm{A}_{2}, \mathrm{~B}_{2}, \mathrm{C}_{2}$, then $\Delta \mathrm{A}_{2} \mathrm{~B}_{2} \mathrm{C}_{2}$ is also an acute triangle. Extend the altitudes of $\Delta \mathrm{A}_{2} \mathrm{~B}_{2} \mathrm{C}_{2}$ to intersect the circumcircle at $\mathrm{A}_{3}, \mathrm{~B}_{3}, \mathrm{C}_{3}$, then $\Delta \mathrm{A}_{3} \mathrm{~B}_{3} \mathrm{C}_{3}$ is still an acute triangle, and so on. Continue this process until obtaining the acute $\Delta A_{n+1} B_{n+1} C_{n+1}$. If the three interior angles of $\Delta \mathrm{A}_{1} \mathrm{~B}_{1} \mathrm{C}_{1}$ are $\alpha, \beta, \gamma$, find the three interior angles of $\Delta A_{n} B_{n} C_{n}$.
The right figure shows the inscribed $\Delta \mathrm{A}_{n} \mathrm{B}_{n} \mathrm{C}_{n}$ and $\Delta \mathrm{A}_{n+1} B_{n+1} C_{n+1}$. Now we use $\angle \mathrm{A}_{n}$ to represent $\angle \mathrm{A}_{a+1}$. Circle
$$
\begin{aligned}
& \angle A_{n+1}=\angle B_{n+1} A_{n+1} C_{n+1} \\
= & \angle B_{n+1} A_{n+1} A_{n}+\angle A_{n} A_{n+1} C_{n+1} \\
= & \angle B_{n+1} B_{n} A_{n}+\angle A_{n} C_{n} C_{n+1} \\
= & \frac{\pi}{2}-\angle A_{n}+\left(\frac{\pi}{2}-\angle A_{n}\right) \\
= & \pi-2 \angle A_{n}, \text { i.e., } \\
& \angle A_{n+1}=-2 \angle A_{n}+\pi .
\end{aligned}
$$

From this, we can see that $\angle \mathrm{A}_{1}, \angle \mathrm{A}_{2}, \cdots, \angle \mathrm{A}_{n+1}$ form a geometric sequence. According to formula (2), we get
$$
\angle \mathrm{A}_{\mathbf{n + 1}}=(-2)^{\mathrm{n}}\left(\alpha-\frac{\pi}{3}\right)+\frac{\pi}{3} .
$$

Similarly,
$$
\begin{array}{l}
\angle \mathrm{B}_{\mathrm{n}+1}=(-2)^{\mathrm{n}}\left(\beta-\frac{\pi}{3}\right)+\frac{\pi}{3}, \\
\angle \mathrm{C}_{\mathrm{n}+1}=(-2)^{\mathrm{n}}\left(\gamma-\frac{\pi}{3}\right)+\frac{\pi}{3} .
\end{array}
$$

This gives the three interior angles of $\Delta A_{n+1} B_{n+1} C_{n+1}$.
This article has benefited from the help of Ms. Wang Yajuan from Beijing Normal University, and the author would like to express gratitude.
(Author's affiliation: Beijing No. 46 High School)