Prove that the tangent line is given by $\frac{\partial F}{\partial X}(P)X+\frac{\partial F}{\partial Y}(P)Y+\frac{\partial F}{\partial Z}(P)Z=0$

Question

This is problem A.$5$ in Rational Points on Elliptic Curves:

Let $C: F(X, Y, Z) = 0$ be a projective curve given by a homogeneous polynomial $F \in \mathbb C[X, Y, Z]$, and let $P \in \mathbb P^2$ be a point.

If $P$ is a non-singular point of $C$, prove that the tangent line to $C$ at $P$ is given by the equation $$\frac{\partial F}{\partial X}(P) X+\frac{\partial F}{\partial Y}(P) Y+\frac{\partial F}{\partial Z}(P) Z=0.$$

I have already proved that

1. the three partial derivatives of $F$ are homogeneous polynomials of degree $d-1$
2. $X \frac{\partial F}{\partial X}+Y \frac{\partial F}{\partial Y}+Z \frac{\partial F}{\partial Z} = d \cdot F(X, Y, Z)$
3. $P$ is a singular point of $C$ if and only if $\frac{\partial F}{\partial X}(P)=\frac{\partial F}{\partial Y}(P)=\frac{\partial F}{\partial Z}(P)=0$.

I think these results are enough to prove it, but I can't see how to write it formally. Anyone got any hints?