KöMaL 2026 B5545

The sides of an acute triangle have lengths $a$, $b$ and $c$. The lengths of the line segments connecting the triangle's isogonal point to its vertices are $x$, $y$ and $z$. Prove that $a^2$, $b^2$ and $c^2$ form an arithmetic sequence (in some order) if and only if $x$, $y$ and $z$ form an arithmetic sequence in some order.