推导(A+B+C)^2=3(A^2+B^2+C^2),则A=B=C

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推导(A+B+C)^2=3(A^2+B^2+C^2),则A=B=C

推导(A+B+C)^2=3(A^2+B^2+C^2),则A=B=C
推导(A+B+C)^2=3(A^2+B^2+C^2),则A=B=C

推导(A+B+C)^2=3(A^2+B^2+C^2),则A=B=C
(a+b+c)^2=3(a^2+b^2+c^2)
2a^2+2b^2+2c^2-2ab-2bc-2ca=0
(a^2-2ab+b^2)+(b^2-2bc+c^2)+(c^2-2ca+a^2)=0
(a-b)^2+(b-c)^2+(c-a)^2=0
所以(a-b)^2=0,(b-c)^2=0,(c-a)^2=0
a-b=0,b-c=0,c-a=0
所以a=b=c

3(A^2+B^2+C^2)-(A+B+C)^2
=(A-B)^2+(B-C)^2+(A-C)^2=0
所以A-B=B-C=A-C=0
所以A=B=C

左边展开-右边=0
=a^2+b^2+c^2+2ab+2ac+2bc-3(a^2+b^2+c^2)
=-(a-b)^2-(b-c)^2-(a-c)^2

(A+B+C)^2=3(A^2+B^2+C^2),
A^2+B^2+C^2+2AB+2BC+2AC=3A^2+3B^2+3C^2
2A^2+2B^2+2C^2-2AB-2BC-2AC=0
(A-B)^2+(A-C)^2+(B-C)^2=0
A=B=C