I didn't bother specifying the correct one because I didn't think it was important enough to the point to be worth the effort to describe. But for completeness, if we use # to denote the operation described by fermigier above, then the group law a+b is given by a+b=(a#b)#O, where O here denotes the vertical point at infinity.
...and at this point we get into a whole can of worms, because that's right this whole time this was actually all taking place in the projective plane, not the affine plane, a complication the article didn't get into, meaning there's this hidden point you didn't know about. And actually we could have used any point as the basepoint and gotten a group law (although they all end up being isomorphic!), which is why technically an elliptic curve is (despite the name) defined to be not just a curve of genus 1, but rather a curve of genus 1 together with a choice of basepoint; the use of the vertical point at infinity as basepoint is just the default convention when you're doing things in this equational way rather than more abstractly, etc... and now you see why I didn't want to get into it.
Also, # isn't a group law because it doesn't satisfy the requirements of one. For instance, there's no identity (maybe barring some weird degenerate cases? I'm not an algebraic geometer so I'm not too familiar with the details here). For # to have an identity P, all of the curve's tangent lines would have to pass through P (and P would have to be an inflection point). Again, not an algebraic geometer but I think that's impossible! (It certainly isn't typical.) I also don't think # is associative but I don't really want to check that right now. Regardless it definitely is not typically a group structure.
