• : a2−ab+b2=(a2−ab+14b2)+34b2=(a−12b)2+34b2a squared minus a b plus b squared equals open paren a squared minus a b plus 1 over 4 end-fraction b squared close paren plus 3 over 4 end-fraction b squared equals open paren a minus 1 over 2 end-fraction b close paren squared plus 3 over 4 end-fraction b squared𝑎2−𝑎𝑏+𝑏2=(𝑎2−𝑎𝑏+14𝑏2)+34𝑏2=(𝑎−12𝑏)2+34𝑏2.
  • Vì (a−12b)2≥0open paren a minus 1 over 2 end-fraction b close paren squared is greater than or equal to 0(𝑎−12𝑏)2≥0và 34b2≥03 over 4 end-fraction b squared is greater than or equal to 034𝑏2≥0với mọi a,ba comma b𝑎,𝑏thực, nên (a−12b)2+34b2≥0open paren a minus 1 over 2 end-fraction b close paren squared plus 3 over 4 end-fraction b squared is greater than or equal to 0(𝑎−12𝑏)2+34𝑏2≥0.
  • Dấu đẳng thức xảy ra khi: (a−12b)2=0open paren a minus 1 over 2 end-fraction b close paren squared equals 0(𝑎−12𝑏)2=0và 34b2=03 over 4 end-fraction b squared equals 034𝑏2=0, tức là a−12b=0a minus 1 over 2 end-fraction b equals 0𝑎−12𝑏=0và b=0b equals 0𝑏=0. Suy ra a=0a equals 0𝑎=0và b=0b equals 0𝑏=0

áp dụng tính chất tỉ lệ thức có: (y+z-x)/x + (z+x-y)/y + (x+y-z)/z= (y+z-x+z+x-y+x+y-z)/(x+y+z)= (x+y+z)/(x+y+z)=1 =>y+z-x=x ; z+x-y=y và x+y-z=z Do đó ta có: (1 + x/y)= [(z+x-y)/y + (y+z-x)/y] =2z/y Tương tự có: 1 + y/z=2x/z và 1 + z/x =2y/x Do đó biểu thức sẽ bằng : 2x/z . 2y/x . 2z/y = 8xyz/xyz =8