Let $a,b,c$ be the sides of a triangle , then what is the maximum and minimum values (if exist) of the following quantities
(i) $\dfrac {a^2b^2c^2}{(a+2b)(a+2c)(b+2c)(b+2a)(c+2a)(c+2b)}$
(ii) $\dfrac {abc(a+b+c)^3}{(a+2b)(a+2c)(b+2c)(b+2a)(c+2a)(c+2b)}$
(i) \begin{align*} &\dfrac {a^2b^2c^2}{(a+2b)(a+2c)(b+2c)(b+2a)(c+2a)(c+2b)}\\ =&\dfrac{1}{(1+2\frac{b}{a})(1+2\frac{a}{b})(1+2\frac{c}{a})(1+2\frac{a}{c})(1+2\frac{c}{b})(1+2\frac{b}{c})} \end{align*} For each two factors, for example, $(1+2\frac{b}{a})(1+2\frac{a}{b})=5+2(\frac{a}{b}+\frac{b}{a})\geqslant5+2\cdot2$
(ii) \begin{align*} &\dfrac {abc(a+b+c)^3}{(a+2b)(a+2c)(b+2c)(b+2a)(c+2a)(c+2b)}\\ =&\frac{(a+b+c)^3}{(1+2\frac{b}{a})(1+2\frac{c}{b})(1+2\frac{a}{c})(a+2c)(b+2a)(c+2b)} \end{align*} Note that $(1+2\frac{b}{a})(1+2\frac{c}{b})(1+2\frac{a}{c})=9+2(\frac{b}{a}+\frac{c}{b}+\frac{a}{c})+4(\frac{c}{a}+\frac{b}{c}+\frac{a}{b})\geqslant9+2\cdot3+4\cdot3$ and $(a+2c)(b+2a)(c+2b)$