Let $f, g \in k[X_0, X_1, X_2]$ be homogeneous polynomials of degrees $d_1$ and $d_2$ respectively. Suppose $V(f)$ and $V(g)$ share no common irreducible component. Then there exist non-negative integers $m_p$ (intersection multiplicities) such that \begin{align*} \sum_{p \in V(f) \cap V(g)} m_p = d_1 d_2. \end{align*} In particular, $|V(f) \cap V(g)| \leq d_1 d_2$, with equality (counted with multiplicity) always holding.