Minimizing in 3 variables 
Find the least possible value of the fraction $\dfrac{a^2+b^2+c^2}{ab+bc}$, where $a,b,c > 0$.

My try:
$a^2+b^2+c^2 = (a+c)^2 - 2ac + b^2$,
$= (a+c)/b  +b/(a+c)  -2ac/b(a+c)$
AM > GM
$3\sqrt[3]{-2ac/b(a+c)}$
And I cant somehow move on.

Write
$$\frac{2a^2+b^2+b^2+2c^2}{2}=\frac{2a^2+b^2}{2}+\frac{b^2+2c^2}{2}\geq \sqrt{2}ab+\sqrt{2}bc$$
