Let $G$ be a circle of radius $R > 0$. Let $G_1, G_2,…,G_n$ be $n$ circles of equal radius $r > 0$, Then prove the following;

Question

Let $G$ be a circle of radius $R > 0$. Let $G_1, G_2, …, G_n$ be $n$ circles of equal radius $r > 0$. Suppose each of the $n$ circles $G_1, G_2, …, G_n$ touches the circle $G$ externally. Also for $i = 1, 2, 3,…,n-1$, the circle $G_i$ touches $G_{i+1}$ externally and $G_n$ touches $G_1$ externally, then prove that if $n=12$, $\sqrt2(\sqrt 3 +1)r>R$.

I started this question with drawing out $G_1$ and $G_2$ and then tried the concept of the center of $G$ being bisected $n$ times, as in the angle subtended by the centers of $G_1$ and $G_2$ on center of $G$ as $\frac{2π}{n}$. It's here that I'm stuck, most probably as there is some trigonometry involved to create a relation with $r$ and $R$. Is this approach the shortest and correct?

EDIT- i managed to get $(r+R)\sin\frac{π}{n}=r$.

Answer 1

Hint : $$R = r( \csc \frac{π}{n} - 1),$$