RO-TST 1996 Problem 14

Let $x,y,z$ be real numbers. Prove that the following conditions are equivalent:

(i) $x,y,z$ are positive numbers and $\dfrac 1x + \dfrac 1y + \dfrac 1z \leq 1$ ; (ii) $a^2x+b^2y+c^2z>d^2$ holds for every quadrilateral with sides $a,b,c,d$ .