I have a projective plane: PG$(2, \mathbb{F})$, where char$\mathbb{F}\neq 2$. I have a C conics and an l line. U is a point in C'=C\l. There is an operation on C'. If A,B $\in$ C' and $H_{AB}$ $= AB \cap l$:
$A*B$ =
\begin{cases} U,& \text{if } H_{AB}U \text{ is tangent to C at U}\\ D, & \text{if} H_{AB}U \cap C = \{D,U\} \end{cases}
I have to show that (C', $*$) is isomorphic to the additive group of $\mathbb{F}$ if l is a tangent to C, or to the multiplicative group of $\mathbb{F}$ if l is a secant to C.
So far I proved that (C', $*$) is an Abelian group. To show that these groups are isomorphic, I think I have to find an isomorphism. I know that $\mathbb{F}^*=\mathbb{F}$\{$0$} is the multiplicative group and it is commutative, and that the additive group is also commutative. I am not sure how to continue. I believe we can focus on the cases where the conic is a parabola or a hyperbola, but I have yet to find the proof of these 2 statements. I would be grateful if you could help me.
First of all, it's easy to prove that $A*U=A$ for any $A\in C'$, hence $U$ is the identity element. Then, let $K$ be the intersection between $l$ ant the tangent to $C$ at $U$: for any $A\in C'$ we can construct $A^{-1}$ as the other intersection of line $AK$ with $C$.
From $K$ a second line tangent to $C$ can be constructed, with tangency point $V$. This line is the same as $l$ if $l$ is tangent to $C$, in which case $V\not\in C'$. But if $l$ is secant to $C$, then $V\in C'$ and $V*V=U$. Hence $V$ plays the role of $-1$ in the multiplicative group.