Let $G$ be a circle of radius $R > 0$. Let $G_1, G_2, \dots, G_n$ be $n$ circles of equal radius $r > 0$. Suppose each of the $n$ circles $G_1, G_2, G_2, \dots, G_n$ touches the circle $G$ externally. Also, for $i = 1, 2, \dots, n - 1$, the circle $G_i$ touches $G_{i+1}$ externally, and $G_n$ touches $G_1$ externally. Then, which of the following statements is/are TRUE? (1) If $n = 4$, then $\left(\sqrt{2} - 1\right) r < R$ (2) If $n = 5$, then $r < R$ (3) If $n = 8$, then $\left(\sqrt{2} - 1\right) r < R$ (4) If $n = 12$, then $\sqrt{2}\left(\sqrt{3} + 1\right) r > R$

Let G be a circle of radius R > 0. Let G₁, G₂, …, Gₙ be n circles of equal radius r > 0. Suppose each of the n circles G₁, G₂, G₂, …, Gₙ touches the circle G externally. Also, for i = 1, 2, …, n - 1, the circle Gᵢ touches Gᵢ₊₁ externally, and Gₙ touches G₁ externally. Then, which of the following statements is/are TRUE? $\text{Given: } G \text{ has radius } R;\quad G_1,\dots,G_n \text{ radius } r$ $\text{each } G_i \text{ touches } G \text{ externally}$ $G_i \text{ touches } G_{i+1} \ (i=1..n{-}1);\quad G_n \text{ touches } G_1$ $\text{To find: which of (1)--(4) are TRUE}$ $(1)\ n=4:\ (\sqrt2-1)r<R \qquad (2)\ n=5:\ r<R$ $(3)\ n=8:\ (\sqrt2-1)r<R \qquad (4)\ n=12:\ \sqrt2(\sqrt3+1)r>R$ $\text{Plan: centres form a regular } n\text{-gon}$ $\text{circumradius } R+r,\quad \text{side } 2r$ $\text{use two adjacent centres } A,B \text{ with } O$ $G:\ \text{centre } O,\ \text{radius } R$ $OG_i = R + r\ \ \forall i\quad (\text{external tangency})$ $\text{centres lie on a circle of radius } R+r$ $OA = OB = R+r,\qquad AB = 2r$ $\angle AOB = \dfrac{2\pi}{n}$ $AB = 2(R+r)\sin\!\left(\dfrac{\pi}{n}\right)$ $2(R+r)\sin\!\left(\dfrac{\pi}{n}\right) = 2r$ $(R+r)\sin\!\left(\dfrac{\pi}{n}\right) = r$ $\dfrac{R}{r} = \dfrac{1}{\sin(\pi/n)} - 1 = \operatorname{cosec}\!\left(\dfrac{\pi}{n}\right) - 1$ $\dfrac{R}{r} = \operatorname{cosec}\!\left(\dfrac{\pi}{n}\right) - 1\ \text{increases with } n$ $n:\ 4\ \ 5\ \ 8\ \ 12 \quad\Longrightarrow\quad \dfrac{R}{r}:\ 0.414\ \ 0.701\ \ 1.613\ \ 2.864$ $n=4:\ \dfrac{R}{r}=\sqrt2-1\ \Rightarrow\ R=(\sqrt2-1)r$ $\text{equality, not } R>(\sqrt2-1)r\ \Rightarrow\ \textbf{FALSE}$ $n=5:\ \dfrac{R}{r}=\operatorname{cosec}36^\circ-1\approx 1.701-1=0.701$ $R\approx 0.701\,r<r\ \Rightarrow\ \textbf{FALSE}$ $n=8:\ \dfrac{R}{r}=\operatorname{cosec}22.5^\circ-1\approx 2.613-1=1.613$ $1.613>\sqrt2-1\approx 0.414\ \Rightarrow\ \textbf{TRUE}$ $n=12:\ \operatorname{cosec}15^\circ=\sqrt2(\sqrt3+1)\approx 3.864$ $R=\left[\sqrt2(\sqrt3+1)-1\right]r\ \Rightarrow\ \sqrt2(\sqrt3+1)r>R\ \Rightarrow\ \textbf{TRUE}$ $\text{TRUE statements: } (3)\ \text{and}\ (4)$