Let $G$ be a circle of radius $R > 0$. Let $G _ { 1 } , G _ { 2 } , \ldots , G _ { n }$ be $n$ circles of equal radius $r > 0$. Suppose each of the $n$ circles $G _ { 1 } , G _ { 2 } , \ldots , G _ { n }$ touches the circle $G$ externally. Also, for $i = 1,2 , \ldots , n - 1$, 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 ? (A) If $n = 4$, then $( \sqrt { 2 } - 1 ) r < R$ (B) If $n = 5$, then $r < R$ (C) If $n = 8$, then $( \sqrt { 2 } - 1 ) r < R$ (D) If $n = 12$, then $\sqrt { 2 } ( \sqrt { 3 } + 1 ) r > R$
Let $G$ be a circle of radius $R > 0$. Let $G _ { 1 } , G _ { 2 } , \ldots , G _ { n }$ be $n$ circles of equal radius $r > 0$. Suppose each of the $n$ circles $G _ { 1 } , G _ { 2 } , \ldots , G _ { n }$ touches the circle $G$ externally. Also, for $i = 1,2 , \ldots , n - 1$, 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 ?
(A) If $n = 4$, then $( \sqrt { 2 } - 1 ) r < R$\\
(B) If $n = 5$, then $r < R$\\
(C) If $n = 8$, then $( \sqrt { 2 } - 1 ) r < R$\\
(D) If $n = 12$, then $\sqrt { 2 } ( \sqrt { 3 } + 1 ) r > R$