Fix $n, m\ge0$ and let $A=\mathbb{K}[T_0,...,T_n]$ and $F,G \in A$ two homogeneous, nonzero, polynomial of the same degree $m$. Show that $$V(F)=V(G) \qquad iff \quad\exists \lambda \neq 0 \quad\text{such that}\quad G=\lambda F$$
Note that for any $F \in A$, $V(F)$ is the closed subset of $\mathbb{P}^n$ (with Zariski topology) defined as the zero set of $F$ in $\mathbb{P}^n$, e.g $V(F)=\{(x_0:\cdots:x_n) \in \mathbb{P}^n \,:\, F(x_0,...,x_n)=0\}$ and called projective hypersurface defined by $F$.
Trivially if $G=\lambda F$ for some $\lambda\neq 0$ then $V(F)=V(G)$.
On the other side, let $F,G \in A$ homogeneous of the same degree $m$ such that $V(F)=V(G)$. By Nullstellensatz theorem this means that $\sqrt{(F)}=\sqrt{(G)}$ and so some power of $F$ divides G and conversely some power of $G$ divides $F$. Is there a way from this to conclude that $F$ and $G$ are proportional? Probably it's such a simple proof but I can't find now any good attempt.
I'm just trying to understand the Veronese embedding has declared here.