Given $\Delta ABC$ and $AE//BC$[...] Prove that: $NP=NQ$

[stanleymitchell]

Given a triangle $ABC$ with orthocenter $F$, altitude $AH$, and median $AD$. A line segment $AE$ is parallel to $BC$ such that $E$ lies on the circumcircle $(O)$ of $\triangle ABC$ and $E \neq A$. Let $AE//BC$($E\in(O)$ and $E\neq A$. $EH\cap(O)=L$;$EH\cap(APH)=Q$, where $P$ is the foot of the perpendicular from $F$ to $AD$.
Prove that: $NQ=NP$, where $N$ is the midpoint of $AL$

Oh, it seems a little bit difficult to upload an image on this forum.

Even though I know how to solve it but I think my auxiliary line is kinda unnatural. And I want other's very first idea to solve this such problem.
Btw, my solution is let $J$ such that $\Delta ACJ$ is acute at $C$ and similar to $\Delta BHF$ and $\Delta AHC$