ab(a+b)+bc(b+c)+ca(c+a)=
a²b+ab²+b²c+bc²+c²a+ca²=
b(a²+c²)+a(b²+c²)+c(b²+a²)=
b(a-c)²+2abc + a(b-c)² +2abc +c(a-b)²+2abc=
b(a-c)²+ a(b-c)² +c(a-b)²+6abc
b(a-c)²≥0
a(b-c)²≥0
c(a-b)²≥0
b(a-c)²+ a(b-c)² +c(a-b)²+6abc≥6abc
ab(a+b)+bc(b+c)+ca(c+a)≥6abc
ab(a+b)+bc(b+c)+ca(c+a)=
a²b+ab²+b²c+bc²+c²a+ca²=
b(a²+c²)+a(b²+c²)+c(b²+a²)=
b(a-c)²+2abc + a(b-c)² +2abc +c(a-b)²+2abc=
b(a-c)²+ a(b-c)² +c(a-b)²+6abc
b(a-c)²≥0
a(b-c)²≥0
c(a-b)²≥0
b(a-c)²+ a(b-c)² +c(a-b)²+6abc≥6abc
ab(a+b)+bc(b+c)+ca(c+a)≥6abc