how to construct the intersection of an hyperbola with a line?

Question

We ONLY have two lines $L$ and $L'$ that intersects in $\mathscr{O}$ and a point $A$ belonging to an affine hyperbola $\mathscr{H}$.

We know that $L$ and $L'$ are the asymptotes of the hyperbola $\mathscr{H}$. We then have a line $\Delta$ and we would like to construct with rule and compass only, the intersection points $B,C$ of $\Delta$ and hyperbola $\mathscr{H}$.

Do you know a way to proceed ? Many thanks.

edit: the intersection points $\mathscr{H} \cap \Delta$ are $B,C$ (not $C,D$).

Answer 1

Someone gave me the answer.

I post his own construction by rule and compass. He is a brilliant geometer ! I give it without any comments if you want to analyze the problem by yourself.

The intersection points we want to construct are $B$ and $C$ (not $C$ and $D$ as I said in the first message).