<s>[INST] Prove that if $a, b, c \in \mathbb{Z^+}$ and $a^2+b^2=c^2$ then ${1\over2}(c-a)(c-b)$ is a perfect square. 
Prove that if $a, b, c \in \mathbb{Z^+}$ and $a^2+b^2=c^2$ then ${1\over2}(c-a)(c-b)$ is a perfect square.

I have tried to solve this question and did pretty well until I reached the end, so I was wondering if I could get help on that part. Here is what I did.
$$a^2+b^2=c^2$$ $$b^2=(c-a)(c+a)$$ Since $a, b, c > 0 \therefore (c+a) \ne 0$ $$\therefore c-a={b^2\over c+a}$$ Similarly we get, $$c-b={a^2\over c+b}$$ $$\therefore {1\over2}(c-a)(c-b)={1\over2}({b^2\over c+a})({a^2\over c+b})$$ $$={(ab)^2\over 2c^2+2ab+2bc+2ca}$$ $$={(ab^2)\over a^2+b^2+c^2+2ab+2bc+2ca}$$ $$={(ab)^2\over (a+b+c)^2}$$ $$=({ab\over a+b+c})^2$$ However, I was unable to prove that ${ab\over a+b+c} \in \mathbb{Z}$ Is there a way to prove it? Thank you
 [/INST] Alternatively, use Formulas for generating Pythagorean triples
WLOG $a=2pqk, b=(p^2-q^2)k,c=(p^2+q^2)k$
$c-a=k(p-q)^2$
$c-b=2kq^2$
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