Left =( aaac+abbb+bccc)/abc is greater than or equal to A+B+C.

Arrange aaac+abbb+bccc greater than or equal to aabc+acbb+abcc.

Aac(a-b)+abb(b-c)+bcc(c-a) is greater than or equal to 0.

Aac(a-b) is greater than or equal to 0, abb(b-c) is greater than or equal to 0, and bcc(c-a) is greater than or equal to 0.

So c is greater than 0, a is greater than 0, and b is greater than 0.

Supplementary note: aa (that is, A times A) must be a positive number, and abc cannot be 0, because they are denominators in the title.