Let us first assume that a + b = c
Then
a^2 + b^2 = c^2
a^2 + b^2 = ( a + b)^2
a^2 + b^2 = a^2 + 2ab + b^2
0 = 2ab which is impossible since a , b are positive
Next, assume that a + b < c
Then.....there must be some positive n such that a + b + n = c
So
a^2 + b^2 = c^2
a^2 + b^2 = (a + b + n)^2
a^2 + b^2 = a^2 + 2ab + 2an + b^2 + 2bn + n^2
0 = 2ab + 2an + 2bn + n^2 which is also impossible since a,b and n are positive
So....
a + b = c is false
a + b < c is false
And the only thing left is that a + b > c
!
