Brilliant asked me for a title

Note that $2(a^2b^2+b^2c^2+c^2a^2)-a^4+b^4+c^4=(a+b+c)(a+b-c)(b+c-a)(c+a-b)=0$ .

WLOG, assume $c=max(a,b,c)=>c=a+b$ .

For a triangle of side lengths $\sqrt{a},\sqrt{b},\sqrt{c}$ to exist, ${ (\sqrt { a } +\sqrt { b } ) }^{ 2 }\ge c=>a+b+2\sqrt { ab } \ge c=>2\sqrt { ab } \ge 0$ , which is obviously true.