Find minimum value of $\frac{cb}{1-c} +\frac{ac}{1-a} + \frac{ba}{1-b}$ How do I minimize the expression
$$f(a,b,c)=\frac{cb}{1-c} +\frac{ac}{1-a} + \frac{ba}{1-b}$$
subject to the constraint $abc=(1-a)(1-b)(1-c)$ with $a$, $b$, $c \in (0,1)$. Conceptually, the Lagrange multiplier procedure can be utilized. But, the resorting algebra seems messy and hard to figure. It is not hard to guest that 3/2 is the minimum given the symmetry. So, I suspect the inequality of certain AM-GM sort could be the answer, yet don't know how to apply. Would appreciate some ideas.

Let $\frac{a}{1-a}=\frac{y}{x},$ $\frac{b}{1-b}=\frac{z}{y},$ where $x$, $y$ and $z$ are positives.
Thus, $\frac{c}{1-c}=\frac{x}{z}$, $a=\frac{y}{x+y}$ and by Nesbitt we obtain:
$$\sum_{cyc}\frac{ab}{1-b}=\sum_{cyc}\left(\frac{y}{x+y}\cdot\frac{z}{y}\right)=\sum_{cyc}\frac{z}{x+y}\geq\frac{3}{2}.$$
The equality occurs for $a=b=c=\frac{1}{2},$ which says that we got a minimal value.
