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?