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

• But you don't even need to show that $\dfrac{ab}{a+b+c}$ is an integer directly. You know that $\dfrac{1}{2}(c-a)(c-b)$ must be a positive integer (since $c$ must have the same parity as $a$ or $b$). The square root of a positive integer is either an integer or an irrational number by the Rational Root Test Theorem. As $\dfrac{ab}{a+b+c}$ is not irrational, it must be an integer. – Batominovski Dec 7 '18 at 8:09

Alternatively, use Formulas for generating Pythagorean triples

WLOG $$a=2pqk, b=(p^2-q^2)k,c=(p^2+q^2)k$$

$$c-a=k(p-q)^2$$

$$c-b=2kq^2$$

Multiply by conjugate: $${ab\over a+b+c}={ab\over a+b+c}\cdot \frac{a+b-c}{a+b-c}=\frac{ab(a+b-c)}{2ab}=\frac{a+b-c}{2}\in \mathbb Z^+,$$ because: $$a+b>c$$ and there are two cases for $$a^2+b^2=c^2$$: 1) $$a,b,c$$ are even; 2) one is even, the other two are odd. And for each case, $$a+b-c$$ is even.