Number of lines tangent to multiple points of a non-singular projective plane curve

Question

Let $C:F(x,y,z)=0$ be a projective plane curve, where $F$ is a homogenous polynomial of degree $d$. I want to show that there exists $p\in \mathcal{P}_2$ such that all lines through $p$ are tangent to $C$ at one point, at the most. I know, for example, that a general quartic plane curve has 28 bitangent lines. I'd like to generalize this to a curve of degree $d$, although I don't need to know the exact number, just that it is finite. Does anyone have any idea of how to approach this proof?