The problem is from a geometry exercise and appears to focus on proving certain properties of a triangle $\triangle ABC$ with given conditions. Let me analyze the content and summarize the tasks:

Given:

1. $\triangle ABC$ is an acute triangle, with $AB < AC$, and is inscribed in a circle $(O)$.
2. $AD, BE,$ and $CF$ are the altitudes, with $H$ as the orthocenter.
3. Line $AD$ intersects $EF$ at $I$.
4. $N$ is the midpoint of $AH$.
5. Line $BI$ intersects $NC$ at $K$, and $CI$ intersects $NB$ at $L$.

Tasks:

(a) Prove $DI \cdot DN = DE \cdot DF$ and that $I$ is the circumcenter of $\triangle NBC$.
(b) Let $HK$ and $HL$ intersect $BC$ at $P$ and $Q$, respectively. Prove that $FP$ and $EQ$ intersect on $(O)$.
(c) Prove that $AD, EP,$ and $FQ$ are concurrent.

Would you like me to solve any specific part (a, b, or c) or provide step-by-step guidance for the entire problem?