Geometric reason why elliptic curve group law is associative

Question

The question title says it all.

I am looking for a geometric proof for the fact that the group law defined on elliptic curves is associative.

I've heard somewhere about something on the internet about 2 cubics intersecting at 8 points must have a ninth point in common but I've never understood what that meant. Maybe this is what I'm looking for.

If a well-known reference has this result and I could easily find it in a library (or better yet over the internet, which I did not manage to find), then I'd accept this as an answer too.

Thanks in advance,

Answer 1

This is contained in any intro book on elliptic curves. Here are two ones which are accessible to undergraduates:

1. McKean and Moll, "Elliptic Curves: Function Theory, Geometry, Arithmetic".
2. Silverman and Tate, "Rational points on elliptic curves"

Answer 2

There is a geometric proof of associativity in the elementary undergraduate book by Silverman and Tate Rational Points on Elliptic Curves.
The proof there is indeed along the lines you suggest of considering a pencil of cubics with nine base points, and is illustrated by a nice drawing.
The textbook derives from 1961 lectures by Tate, one of the best specialists ever in elliptic curves (he received the prestigious Abel prize in 2010).