Subject: Mathematics — Circles
Question: 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 $G_{1}$, $G_{2},G_{2}\ldots,G_{n}$
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. If $n = 12$, find the relation in $r$ & $R$.
Answer: if $n = 12$, then $\sqrt{2}(\sqrt{3} + 1)r > R$
Solution: Refer to diagram,
[IMAGE] In $\bigtriangleup AOB$\
$${sin\left( \frac{\pi}{n} \right) = \frac{r}{R + r} \Rightarrow cosec\left( \frac{\pi}{n} \right) = \frac{R}{r} + 1
}{\Rightarrow R = r\left\lbrack cosec\left( \frac{\pi}{n} \right) - 1 \right\rbrack
}$$If $n = 12$, then $R = r\left( cosec\frac{\pi}{12} - 1 \right)$\
$${R = r(\sqrt{2}(\sqrt{3} + 1) - 1)
}{R < \sqrt{2}(\sqrt{3} + 1)r}$$