Let $F$, $G$, and $H$ be linear functionals on a real vector space $V$. Assume that there is $x \in V$ such that $G(x)<0$ and $F(x) \geq 0$.
Assume further that $F$, $G$, $H$ satisfy the following condition: for any $x,y \in V$, if $G(x)<0$, $G(y)<0$, and $F(x)/G(x)=F(y)/G(y) \leq 0$, then $H(x) \geq 0$ iff $H(y) \geq 0$.
Does this imply that $H=aF+bG$ for some constants $a$ and $b$?
By assumption, there is $x \in V$ such that $G(x) < 0$ and $F(x) \geq 0$. Pick $y \in V$ such that $F(y)=G(y)=0$. Then $F(\lambda x)/G(\lambda x) = F(\lambda x \pm y)/G(\lambda x \pm y) \leq 0$ for any $\lambda >0$. By the condition, we must have $H(\lambda x) \geq 0$ iff $H(\lambda x \pm y) \geq 0$. This relation holds for any $\lambda >0$ if and only if $F(y)=0$. Now the required result follows from Intersection of kernels and linear dependence of functionals.