Question

已知实数a、b、c满足a²+b²+c²-ab-bc-ac=0,求a,b,c的关系（求过程）

Answer 1

a^2+b^2+c^2-ab-bc-ca=0
两端乘2
2a^2+2b^2+2c^2-2ab-2bc-2ca=0
a^2 - 2ab + b^2 + b^2 -2bc + c^2 + c^2 -2ac + a^2 = 0
(a-b)^2 + (b-c)^2 + (c-a)^2 = 0
a-b = 0
b-c = 0
c-a = 0
所以 a = b =c

Answer 2

a²+b²+c²-ab-bc-ac=0
2a²+2b²+2c²-2ab-2bc-2ac=0
a²+b²-2ab+b²+c²-2bc+a²+c²-2ac=0
(a-b)²+(b-c)²+(a-c)²=0
(a-b)²=0,(b-c)²=0,(a-c)²=0
a=b,b=c,a=c
所以a=b=c

Answer 3

a2+b2大于等于2ab，（均值定理）当且仅当a=b时等号成立，同理可证b2 c2,c2 a2
所以a=b=c