TITLE: Geometric reason why elliptic curve group law is associative
QUESTION [14 upvotes]: 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,

REPLY [10 votes]: 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).

REPLY [4 votes]: This is contained in any intro book on elliptic curves.  Here are two ones which are accessible to undergraduates:

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