VT=ab(a+b-2c)+bc(b+c-2a)+ca(c+a-2b)
=a^2*b+a*b^2-2abc+b^2*c+b*c^2-2abc+c^2*a+c*a^2-2abc
=(a^2*b-abc)+(a*b^2-abc)+(b^2*c-abc)+(b*c^2-abc)+(c*a^2-abc)+(c^2*a-abc)
=ab(a-c)+ab(b-c)+bc(b-a)+bc(c-a)+ca(a-b)+ac(c-b)
=(ab(a-c)+bc(c-a))+(ab(b-c)+ac(c-b))+(bc(b-a)+ca(a-b))
=(ab(a-c)-bc(a-c))+(ab(b-c)-ac(b-c))+(bc(b-a)-ca(b-a))
=(a-c)*(ab-bc)+(b-c)*(ab-ac)+(b-a)*(bc-ca)
=(a-c)*b*(a-c)+(b-c)*a*(b-c)+(b-a)*c*(b-a)
=(a-c)^2*b+(b-c)^2*a +(b-a)^2*c
Ta có: (a-c)^2>=0 ;a>0(vì a là cạnh của tam giác)
=>(a-c)^2*a>=0
CMTT (b-c)^2*a>=0
(b-a)^2*c>=0
=>(a-c)^2*a+(b-c)^2*a+(b-a)^2*c>=0
(đpcm)
dấu = xảy ra khi a=b=c