已知非零实数a,b,c满足6(a^2+b^2+c^2)=(a+b+2c)^2
问题描述：
已知非零实数a,b,c满足6(a^2+b^2+c^2)=(a+b+2c)^2
下午5点前要,满意再加财富
求a:b:c
答
6(a^2+b^2+c^2)=(a+b+2c)^2
6a^2+ 6b^2 + 6c^2 = a^2 + b^2 + 4c^2 + 2ab +4bc +4ac
5a^2 + 5b^2 + 2c^2 -2ab -4bc -4ac = 0
(a^2-2ab +b^2) + (4a^2- 4ac + c^2) + (4b^2 - 4bc + c^2) = 0
(a-b)^2 + (2a-c)^2 + (2b-c)^2 = 0
因为(a-b)^2 >=0,(2a-c)^2>=0 ,(2b-c)^2>=0
所以 (a-b)^2 =0,(2a-c)^2=0 ,(2b-c)^2=0
即 a - b =0 ,2a - c =0,2b - c = 0
所以 a = b = 1/2c
a :b :c = 1:1:2