TITLE: Why do we define the group law on elliptic curves only for Weierstrass forms and $O$ an inflexion point?
QUESTION [8 upvotes]: In almost all texts concerning the group law on an elliptic curve it is first proven that any nonsingular cubic can be given by a Weierstrass equation and then the group law using the point $O$ at infinity is described (which in this case is an inflexion point).
Since I seem to be able to geometrically define addition of points $P$ and $Q$ for any nonsingular cubic and any point $O$ on the curve chosen as the identity element, I wonder why this is not mentioned more often. 
I see that $P+Q+R=O$ for any collinear points $P$, $Q$ and $R$ only if $O$ is an inflexion point. Of course it all gets much easier with this property at hand, but is it 'needed'?
Maybe I did miss some counterexample, but isn't it possible to have a nice geometric addition law for all sorts of elliptic curves, not only for the ones in Weierstrass form with $O$ the (inflection) point at infinity?
Thank you for your insights.
Edit: Probably the title was a bit misleading. The more basic thing to ask seems to be Why do we require $P+Q+R=O$ for collinear points?

REPLY [9 votes]: If you have any genus one curve $E$, any point $O$ on $E$, then there is a unique algebraic group structure on $E$ making $O$ the identity element.  This applies in particular when $E$ is a smooth plane cubic.  
However, it is not hard to show that given $E$ and $O$, we can always embed $E$ as a plane cubic in such a way that $O$ is furthermore an inflection point of $E$.
Since the discussion of the group law is a little simpler when $O$ is an inflection point (and indeed becomes simplest when $O$ is moved to be a point at infinity, with its tangent line being the line at infinity), most books put themselves in that case when discussing the group law.