若(a+b+c)^2=3ab+3bc+3ac.求证a=b=c.

4个回答

  • (a + b + c)² = 3ab + 3bc + 3ac

    a² + b² + c² + 2ab + 2bc + 2ac = 3ab + 3bc + 3ac

    a² + b² + c² - ab - bc - ac = 0

    2a² + 2b² + 2c² - 2ab - 2bc - 2ac = 0

    a² - 2ab + b² + b² - 2bc + c² + a² - 2ac + c² = 0

    (a - b)² + (b - c)² + (a - c)² = 0

    因为一个数的平方大于等于0

    所以只有当 a - b = 0 且 b - c = 0 且 a - c = 0 时等号成立

    所以 a = b = c