Let $C$ a non singular cubic projective plane curve with a fix point $O$, we use que chord-tangent method to make $G = (C,+,O)$ an abelian group, if I choose another fix point $O'$ and construct the abelian group $H=(C , + ,O')$ in the same way, then $G\cong H$ ?
I guess that this is very intuitive, but I don't know how to prove it.
HINT.-In the figure you have the point $-(A + B)$ it is always such, regardless of which is the zero of the group. There is also in this figure two sums $A + B$ of the same points $A$ and $B$, corresponding to the standard zero at the point of infinity and other marked with the subscript $1$ corresponding to the zero point $O$.
It's better than you try to finish and if you can not, I will return.