Square of rational

Let $x, y, z \in \mathbb{R}^*$ such that it holds $x+y+z=0$. Prove that the number

$$\mathcal{A}= \frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}$$

is a square of rational number.

Solution

Indeed we have successively:

$$\begin{aligned}
\frac{1}{x^2}+ \frac{1}{y^2}+ \frac{1}{z^2} &=\frac{x^2y^2 + y^2z^2+z^2x^2}{x^2y^2z^2} \\
 &= \frac{\left ( xy+yz+zx \right )^2- \cancelto{0}{2xyz \left ( x+y+z \right )}}{x^2y^2z^2}\\
 &= \left ( \frac{xy+yz+zx}{xyz} \right )^2
\end{aligned}$$

ending the exercise.