MathNet 2010 — Part 02 Question 191

Let $x$, $y$ and $z$ be positive integers satisfying $\text{gcd}(x, y, z) = 1$. Prove that if $(y^2 - x^2) - (z^2 - y^2) = ((y - x) - (z - y))^2$, then $x$ and $z$ are perfect squares.