Question 91
Consider positive integers $a, b, c$ that are side-lengths of a non-degenerate triangle and such that $\text{GCD}(a, b, c) = 1$ and the fractions
$$
\frac{a^2 + b^2 - c^2}{a + b - c}, \quad \frac{b^2 + c^2 - a^2}{b + c - a}, \quad \frac{c^2 + a^2 - b^2}{c + a - b}
$$
are all integers. Prove that the product of the denominators of the three fractions is either a square or twice a square of an integer. (Jaromír Šimša)
$$
\frac{a^2 + b^2 - c^2}{a + b - c}, \quad \frac{b^2 + c^2 - a^2}{b + c - a}, \quad \frac{c^2 + a^2 - b^2}{c + a - b}
$$
are all integers. Prove that the product of the denominators of the three fractions is either a square or twice a square of an integer. (Jaromír Šimša)