I think this may be what you are trying to do. where the n can be any number.
f[n_] := Cases[
Tuples[Range[n], {3}], {a_, b_, c_} /;
a <= b <= c && c <= a + b && IntegerQ[Sqrt[2 a^2 + 2 b^2 - c^2]/2]]
f[10]
{{1, 1, 2}, {1, 3, 4}, {1, 5, 6}, {1, 7, 8}, {1, 9, 10}, {2,
2, 4}, {2, 4, 6}, {2, 6, 8}, {2, 8, 10}, {3, 3, 6}, {3, 5, 8}, {3,
7, 10}, {4, 4, 8}, {4, 6, 10}, {5, 5, 6}, {5, 5, 8}, {5, 5, 10}, {6,
8, 10}}
Paul.
I noticed that some of those results were zero, so if we want to test for the Integer part to be greater than zero we can add another test like so
f[n_] := Cases[
Tuples[Range[n], {3}], {a_, b_, c_} /;
a <= b <= c && c <= a + b && c > 0 &&
IntegerQ[p = Sqrt[2 a^2 + 2 b^2 - c^2]/2] && p > 0]
Notice I set the sqrt part to equal p and then tested the p to be greater than zero afterwards.
P