Question

已知a平方+b平方+c平方=ab+ac+bc,求证a=b=c

Answer 1

a^2+b^2+c^2=ab+ac+bc

两边同乘以2

2a^2+2b^2+2c^2=2ab+2ac+2bc

整理得

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

(a-b)^2+(b-c)^2+(a-c)^2=0

故

(a-b)^2=0

(b-c)^2=0

(a-c)^2=0

故

a-b=0

b-c=0

a-c=0

故

a=b=c