KöMaL 2020 B5103

Let $a$, $b$, $c$, $x$, $y$ and $z$ be positive numbers that satisfy the equalities $a^2+b^2=c^2$ and $x^2+y^2=z^2$. Prove that ${(a+x)}^2+{(b+y)}^2\le {(c+z)}^2$, and determine the condition for equality.