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*cTa có: (a-c)^2>=0 ;a>0(vì a là cạnh của tam giác) =>(a-c)^2*a>=0CMTT (b-c)^2*a>=0 (b-a)^2*c>=0 =>(a-c)^2*a+(b-c)^2*a+(b-a)^2*c>=0(đpcm)
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*cTa có: (a-c)^2>=0 ;a>0(vì a là cạnh của tam giác) =>(a-c)^2*a>=0CMTT (b-c)^2*a>=0 (b-a)^2*c>=0 =(a-c)^2*a+(b-c)^2*a+(b-a)^2*c>=0(đpcm)
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*cTa có: (a-c)^2>=0 ;a>0(vì a là cạnh của tam giác) =>(a-c)^2*a>=0CMTT (b-c)^2*a>=0 (b-a)^2*c>=0 =
>(a-c)^2*a+(b-c)^2*a+(b-a)^2*c>=0(đpcm)