I'm faced with what seems a paradox. If we have an elliptic curve $E/\Bbb C$ in Weierstrass form so that $\mathcal O_E$ is at infinity, then the addition law is quite easy to picture geometrically. In particular, the 2-torsion points are exactly the ones whose tangent is the vertical line $y=c$.
The 3-torsion points must therefore not have a vertical tangent line (except for $\mathcal O_E$), but instead they must satisfy $P=-P$. Denoting $P=(P_x,P_y)$, this means that $-P=(P_x,-P_y)$. Now my problem is, if the $x$-coordinate of $P$ is going to be different than $P$'s $x$-coordinate ($P_x\neq P_x$) because the tangent line at $P$ is not vertical, how can the secant joining $P$ and $P$ meet $\mathcal O_E$? The last condition implies that $P_x=P_x$, right?