Geometric reason why elliptic curve group law is associative

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,

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).

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"