Question 22
Prove that positive $a$, $b$, $c$ are lengths of sides of a triangle if and only if a system of equations
$$
a(yz + x) = b(zx + y) = c(xy + z), \quad x + y + z = 1
$$
with unknowns $x$, $y$, $z$ has a solution in positive reals.
$$
a(yz + x) = b(zx + y) = c(xy + z), \quad x + y + z = 1
$$
with unknowns $x$, $y$, $z$ has a solution in positive reals.