Angle Bisectors are Harmonic Conjugates
Then $\LL_3$ and $\LL_4$ are harmonic conjugates with respect to $\LL_1$ and $\LL_2$.
From Bisectors of Adjacent Angles between Straight Lines Meeting at Point are Perpendicular, $\LL_3$ is perpendicular to $\LL_4$.
The triangle $\triangle OLM$ has:
- $\angle NOL = \angle NOM$
- $\angle ONL = \angle ONM$ as both are right angles
- $ON$ common
So $\triangle ONL$ and $\triangle ONM$ are congruent.
So $N$ is the midpoint of $LM$.
Hence from Straight Line which cuts Harmonic Pencil forms Harmonic Range, the straight lines $\LL_1$, $\LL_2$, $\LL_3$ and $\LL_4$ form a harmonic pencil.
That is: $\LL_3$ and $\LL_4$ are harmonic conjugates with respect to $\LL_1$ and $\LL_2$.