I am reading Number Theory 1: Fermat's Dream by Kato. In Chapter 1 he defines the group structure on a general elliptic curve
$$y^{2} = ax^{3} + bx^{2} + cx + d$$
(where $a \neq 0$, and the cubic polynomial on the right has no root of multiplicity greater than $1$) by:
1) If $P, Q \in E(K)$ are points on the curve (where $K$ is a field), then $P+Q$ is the unique point $T = (x, -y)$ where $R = (x, y)$ is a third point of intersection of the line passing through $P$ and $Q$.
2) If $P=(a, b) \in E(K)$ then define $-P = (a, -b)$.
3) The point at infinity $O$ acts as the zero element. Not too hard to see from (1) that $P+O = P$ and $P + (-P) = O$.
I get (1) - (3) and can pretty easily show most of the group axioms from this (with the exception of associativity). And the cool thing is that I can actually see what's going on from drawing the curve. What I don't get is how Kato defines the addition of a point with itself (i.e. $P+P = 2P$). He shows this algebraically, which is incredibly messy and has almost no geometry behind it. I'm wondering if there is an easier way to see how this works. (I apologize for leaving out Kato's description of $P+P$, but it is very long and messy).
Geometrically, adding two points on the elliptic curve involves the line through those two points. Algebraically, we want a line such that the curve and line agree on those two points, equivalently the difference between the curve and line have those two points as roots.
Thus adding a point to itself geometrically involves the tangent to the curve at that point, and algebraically involves the line such that the difference between the curve and the line has that point as a double root.