Points of intersection between tangent line and elliptic curve

Question

Let $C$ be an elliptic curve, i.e. a smooth projective curve of degree 3, and $P\in C$. The tangent line in $P$ could intersects the curve $C$ in three distinct points?

Answer 1

It cannot. The curve has degree three, so any line meets the curve in three points, counting multiplicity. Since it's a tangent line, it meets $C$ at $P$ with multiplicity 2, so it can only meet $C$ at one other point.