IrMO 2016 Problem 8

Suppose $a, b, c$ are real numbers such that $abc \ne 0$ . Determine $x, y, z$ in terms of $a, b, c$ such that $bz + cy = a, cx + az = b, ay + bx = c$ . Prove also that $\frac{1 - x^2}{a^2} = \frac{1 - y^2}{b^2} = \frac{1 - z^2}{c^2}$ .