Question

推导：若（a+b+c)²=3（a²+b²+c²）,则a=b=c

Answer 1

证明：（A+B+C)^2=3(A^2+B^2+C^2)

3A^2+3B^2+3C^2=A^2+B^2+C^2+2AB+2AC+2BC

2A^2+2B^2+2C^2-2AC-2AB-2BC=0

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

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

A-B=0,A-C=0,B-C=0

所以A=B=C