Proving a well-known inequality using S.O.S

Let $c=\min\{a,b,c\}$. Hence, $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}-3=\frac{a}{b}+\frac{b}{a}-2+\frac{b}{c}+\frac{c}{a}-\frac{b}{a}-1=\frac{(a-b)^2}{ab}+\frac{(c-a)(c-b)}{ac}\geq0$$

From here we can get a SOS form: $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}-3=\frac{1}{6abc}\sum_{cyc}(a-b)^2(3c+a-b)$$


Here is another SOS (Shortest) $$ab^2+bc^2+ca^2-3abc={\frac {a \left( c-a \right) ^{2} \left({b}^{2}+ ac+cb \right) +b \left( 2\,ab+{c}^{2}-3\,ac \right) ^{2}}{4\,ab+ \left( c-a \right) ^ {2}}} \geqslant 0$$