a^4+b^4+c^4=a^2b^2+b^2c^2+c^2a^2
a^4+b^4+c^4-a^2b^2-b^2c^2-c^2a^2=0
2a^4+2b^4+2c^4-2a^2b^2-2b^2c^2-2c^2a^2=0
(a^4-2a^2b^2+b^4)+(b^4-2b^2c^2+c^4)+(c^4-2c^2a^2+a^4)=0
(a^2-b^2)^2+(b^2-c^2)^2+(c^2-a^2)^2=0
a^2-b^2=0 a=b
b^2-c^2=0 b=c
c^2-a^2=0 c=a
a=b=c
三角形为等边三角形.