证明：由海伦公式,
S =√ [ p (p-a) (p-b) (p-c) ],
其中 p =(a+b+c)/2.
所以 16(S^2) =(a+b+c) (b+c-a) (a+c-b) (a+b-c)
=[ (b+c)^2 -a^2 ] [ a^2 -(c-b)^2 ]
= - [ a^2 -(b+c)^2 ] [ a^2 -(b-c)^2 ]
= -[ (a^4) -2 (a^2) (b^2 +c^2) -(b^2 -c^2)^2 ]
= -(a^4) -(b^4) -(c^4) +2(a^2)(b^2) +2(b^2)(c^2) +2(b^2)(c^2),
所以-(a^4) -(b^4) -(c^4) +2(a^2)(b^2) +2(b^2)(c^2) +2(b^2)(c^2) >0,
即 (a^4) +(b^4) +(c^4) -2(a^2)(b^2) -2(b^2)(c^2) -2(b^2)(c^2)