Consider a circle of radius 1. Describe a regular $n$-sided polygon around it and inscribe a regular $n$-sided polygon in it. Denote their perimeters by $P_{n}$ (for the circumscribed) and $p_{n}$ (for the inscribed).

a) Find $P_{4}, p_{4}, P_{6}$, and $p_{6}$.

b) Prove that the following recurrence relations hold: $\quad P_{2 n}=\frac{2 P_{n} p_{n}}{P_{n}+p_{n}}, \quad p_{2 n}=\sqrt{p_{n} P_{2 n}}$ $(n \geq 3)$.

c) Find $P_{96}$ and $p_{96}$. Prove the inequalities $3^{10} /{ }_{71}<\pi<3^{1 /} /$.
b) Let $a_{n}$ and $b_{n}$ denote the sides of the inscribed and circumscribed regular $n$-gons. The midpoints $K, L, M$ of the sides $A B, B C, C D$ of the circumscribed regular $2 n$-gon are consecutive vertices of the inscribed regular $2 n$-gon. Moreover, $K M$ is a side of the inscribed regular $n$-gon, the point $E$ of intersection of the lines $A B$ and $C D$ is a vertex of the circumscribed regular $n$-gon, and $K$ and $M$ are midpoints of its sides.

![](https://cdn.mathpix.com/cropped/2024_05_06_d3cd76ad0dc53ae543bdg-38.jpg?height=489&width=1334&top_left_y=1019&top_left_x=362)

From the obvious similarity of triangles $K E M$ and $B E C$, we get $\frac{b_{2 n}}{a_{n}}=\frac{b_{n}-b_{2 n}}{b_{n}}$, which simplifies to $b_{2 n}=\frac{a_{n} b_{n}}{a_{n}+b_{n}}$. From the similarity of triangles $K B L$ and $K L M$, we get $\frac{a_{2 n}}{a_{n}}=\frac{b_{2 n}}{2 a_{2 n}}$, which simplifies to $a_{2 n}=\sqrt{\frac{a_{n} b_{2 n}}{2}}$.

Considering that $p_{n}=n a_{n}, P_{n}=n b_{n}$, we obtain the required relations.

c) Using the formulas from b), we get:

$$
\begin{aligned}
& P_{12}=\frac{2 p_{6} P_{6}}{p_{6}+P_{6}}=\frac{48 \sqrt{3}}{6+4 \sqrt{3}}=\frac{24 \sqrt{3}(2 \sqrt{3}-3)}{(2 \sqrt{3})^{2}-3^{2}}=24(2-\sqrt{3}) \\
& p_{12}=\sqrt{p_{6} P_{12}}=\sqrt{6(48-24 \sqrt{3})}=6 \sqrt{2} \cdot \sqrt{4-2 \sqrt{3}}=6 \sqrt{2} \cdot(\sqrt{3}-1)
\end{aligned}
$$

Continuing, we can reach $P_{96} \approx 6.285429, p_{96} \approx 6.282064$ and verify that $6^{20} / 71 \approx 6.281690 < p_{96} < P_{96} < 6^{2} / 7 \approx 6.285714$. It remains to note that $p_{n} < 2 \pi < P_{n}$ for any $n$.

## Answer

a) $P_{4}=8, p_{4}=4 \sqrt{2}, P_{6}=4 \sqrt{3}, p_{6}=6$.

c) $P_{96} \approx 6.285429, p_{96} \approx 6.282064$.