I'm current reading Koblitz's "Introduction to Elliptic Curves and Modular Forms," and the author repeatedly mentions that, given a fixed point $P$, points $Q$ of the form $2Q=P$ are found by taking the lines emanating from $-P$ that are tangent to the curve somewhere, and that there are 4 distinct lines with this property.
I understand that if these 4 lines exist, the resulting points at which they are tangent to the curve are indeed exactly the points $Q$ we are after. What I do not understand is why these 4 distinct lines are always guaranteed to exist.
If you know that an elliptic curve over $\mathbb{C}$ is isomorphic to $S^1 \times S^1$ as a group then you can argue as follows: if $Q$ is any element then there is at least one $P$ such that $2P = Q$. Moreover, if $P_1, P_2$ are two such elements then
$$2(P_1 - P_2) = Q - Q = 0$$
so $P = P_1 - P_2$ satisfies $2P = 0$, and there are exactly four such points, namely $(0, 0), (\frac 12, 0), (0, \frac 12), (\frac 12, \frac 12)$ (thinking of $S^1$ as $\mathbb{R}/\mathbb{Z}$). So $2P = Q$ has $4$ solutions for any $Q$, not just $Q = 0$.