已知(a+b+c)^=3(a^+b^+c^),求a=b=c

来源：百度知道编辑：UC知道时间：2024/05/21 12:30:53

证明:

(a + b + c)^2 = 3 * (a^2 + b^2 + c^2)

a^2 + b^2 + c^2 + 2*ab + 2*ac + 2*bc = 3 * (a^2 + b^2 + c^2)

0 = 2*(a^2 + b^2 + c^2) - 2*ab - 2*ac - 2*bc

(a^2 - 2ab + b^2) + (b^2 - 2bc + c^2) + (a^2 - 2ac + c^2) = 0

(a - b)^2 + (b - c)^2 + (a - c)^2 = 0 (*)
又(a - b)^2 >= 0, (b - c)^2 >= 0, (a - c)^2 >= 0,
根据(*)式(a - b)^2 = 0, (b - c)^2 = 0, (a - c)^2 = 0
所以a = b = c.

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