Question

Why do we call the triplet to( 2m m-1 m+1)pythagorean triples?

Answer 1

let me modify your question, it should be m^2-1, m^2+1 and 2m to be the Pythagorean triplet

for the Pythagorean triplet of three numbers a, b, c they must satisfy the condition of Pythagoras theorem

${a}^{2} + {b}^{2} = {c}^{2}$
Now, if you consider the three sides to be m^2-1, m^2+1 and 2m, and consider a and b to be m^2-1 and 2m respectively,
square and add them,
(m^2-1)^2 + (2m)^2
= (m^2+1)^2,

which satisfies the theorem.

Thus, m^2-1, m^2+1 and 2m form Pythagorean triplet.