Let X and Y be smooth projective plane curves, and P a point of intersection.

I want to show that the intersection multiplicity of P is greater than or equal to 2 if and only if the tangent lines of X and Y at P coincide.

I know that the intersection number of X and Y at P is greater than or equal to the product of their multiplicities. So . Can I use this?

If the intersection multiplicity is great than or equal to 2, it means that at least one of the curves has multiplicity greater than 1 at that point. What does this tell me about the tangent? How do I relate it to the other curve?

Going the other way: if the tangent lines coincide, how do I show that this makes at least one of the curves have multiplicity at P greater than 1?