If (a,b,c) are pythagorean triples such as a^2+b^2=c^2 then prove a^3+b^3+3ab>c^3

Blocked on this for the past hour

I understand they must be positive numbers

Clearly c>a and c>b

but and thinking answer around with using (a+b)^3=a³ + 3a²b + 3ab² + b<sup>3

or</sup> (a+b+c)^3 = a^3+b^3+c^3 + 3(a+b)(b+c)(a +c)

^{but can't find the trick. Anyone can help? Thanks!}

Check your formula - it is incorrect. It works for the Pythagorean triplet 3,4,5:



but not for the triplet 5,12,13:



EDIT - I wonder if the correct formulation of this problem should be to prove that:

?? That would be a breeze (and applies to ALL triangles, not just right angle triangles with integer sides).