Question 619109
Those three expressions form a Pythagorean triple if and only if *[tex \LARGE (m^2 - n^2) + (2mn)^2 = (m^2 + n^2)^2]


(we can show that m^2 + n^2 > 2mn by the AM-GM inequality, m^2 + n^2 > m^2 - n^2 is obvious, given that m,n > 0).


Here, just show that LHS equal the RHS.


*[tex \LARGE (m^4 - 2m^2n^2 + n^4) + 4m^2n^2 = m^4 + 2m^2n^2 + n^4]


This is true, so the three expressions form a Pythagorean triple.