Bisector lines in a 3D space

hello

I am trying to create a bisector line in a 3d space, centered at the intersection of 4 non linear lines. this bisector should be such that both angles are bi sected by the same line. To achieve this i attempted the following grasshopper code. In my code I constructed 2 vectors, working in pairs of 2 lines. I then tried to combine these 2 vectors and get a resultant vector which bisects perfecly but it seems mathematics dont work that way. my “true bisector” is not a bisector.
I would appreciate any help.
perhaps my approach is completely wrong, maybe i need to figure something out that is completely different. Im open to that too

BISEKTOR.gh (12.2 KB)
bisector-attempt.3dm (35.1 KB)

I notice your geometry is very far from the origin. It’s best to avoid this, as it will reduce the accuracy.

As for the bisector, do you mean something like this? - the intersection of the 2 planes which bisect each pair.
bisect.gh (15.5 KB)

yes, this is exactly what i needed! thank you so much. Now i need to implement this sytem into this structure, as in every intersection should be treated as the intersection point in your gh code.
bisector-trial.3dm (634.4 KB)