Let $\mathcal{C}$ be a conic on the projective plane $PG(2,\mathbb{F})$ where $\mathrm{char}\mathbb{F}\neq 2$. Let $\ell$ be a line and let $N$ be a point on $\mathcal{C}\setminus \ell$. For $A,B\in \mathcal{C}$, let $L_{AB}=AB\cap \ell$. Prove that the following operation defines an abelian group $G$ on $\mathcal{C}\setminus\ell$:
$$A \circ B =\begin{cases} N, & \text{ if } L_{AB}N \text{ is the tangent at } N \text{ on }\mathcal{C} \\ M, & \text{ if } L_{AB}N = \{N,M\} \end{cases}.$$
Further show that if
- $\ell$ intersects the conic in one point (tangent) then $G$ is the additive group of $\mathbb{F}$
- $\ell$ intersects the conic in two points (secant) then $G$ is the multiplicative group of $\mathbb{F}$
Proof. Here is a figure where $A\circ B$ equals a point $M$ and $A\circ C$ equals the point $N$.
For the first part I can easily show closure, abelianity, identity (which is $N$) and inverses. I am a bit stuck on the associative part $(A\circ B)\circ C = A\circ (B\circ C)$. Let $A\circ B = M_1$ and $B\circ C = M_2$. I thought we can use Pascal's theorem to show that $M_1C$ is parallel to $AM_2$ but I am not sure how to continue. The second part of the exercise is not clear to me at all. Has anyone seen this type of group construction before? I've been looking at literature and cannot find anything else.
Update: The group part is done, and now we have to consider the cases when $\ell$ is a tangent and when $\ell$ is a secant. I think we can use the parabola and hyperbola for these cases and then use the fact that we can map conics to parabola and hyperbola - we usually do things like this but not sure how to continue.
