If $E(\mathbb{C})$ is an elliptic curve given by $y^2=ax^3+bx+c$ for $a,b,c\in \mathbb{C}$, and $\ell$ is a line tangent to $E(\mathbb{C})$ at some point $p$, then why does $\ell$ meet $E(\mathbb{C})$ at another point that is not $p$? I see that if the tangent line is given by $x=C$ for some constant $C\in \mathbb{C}$, then the line meets $E(\mathbb{C})$ in the point $\mathcal{O}$ at infinity. I also see that since $\mathbb{C}$ is algebraically closed, substituting the equation $y=\alpha x + \beta$ of $\ell$ into $y^2=ax^3+bx+c$ yields three solutions, but why must the third one be different?
EDIT: Of course the tangent at $\mathcal{O}$ meets the curve at only one point, but what about the other points?