 Ask Remember Me? xtiancaz Posts: 1, Reputation: 1 New Member #1 Feb 24, 2017, 05:38 AM
If (a,b,c) are pythagorean triples such as a^2+b^2=c^2 then prove a^3+b^3+3ab&gt;c^3
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 + b3

or
(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! ebaines Posts: 12,130, Reputation: 1307 Expert #2 Feb 25, 2017, 07:28 AM
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).

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