Question

三角形ABC中 a^4+b^4+c^4=a^2b^2+a^2c^2+b^2c^2 怎么证明是等边三角形

Answer 1

a^4+b^4+c^4=a^2b^2+a^2c^2+b^2c^2

2a^4+2b^4+2c^4=2a^2b^2+2a^2c^2+2b^2c^2

(a^4-2a^2b^2+b^4)+(b^4-2b^2c^2+c^4)+c^4-2c^2a^2+c^4)=0

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

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

a^2-b^2=b^2-c^2=c^2-a^2=0

a^2=b^2=c^2

a=b=c