Regarding Pythagorean triples. There are a set of generating formulas. However, if you have two right triangles that are to have the same area, then by the Pythagorean theorem: a^2 + b^2 = c^2 and a'^2 + b'^2 = c'^2. The area of the triangles would be equal, so 1/2*ab = 1/2*a'*b'.
Now if you have one triple, a,b,c, you have ab = a'*b', so what are possible values for a' and b'? Can you find a corresponding c' that is an integer?
Then there is the question: Is a 3-4-5 triangle different from a 4-3-5 triangle?