Let $H$ be the orthocenter of the acute triangle $ABC$. Let $BB'$ and $CC'$ be altitudes of the triangle ($B^{\prime} \in AC$, $C^{\prime} \in AB$). A variable line $\ell$ passing through $H$ intersects the segments $[BC']$ and $[CB']$ in $M$ and $N$. The perpendicular lines of $\ell$ from $M$ and $N$ intersect $BB'$ and $CC'$ in $P$ and $Q$. Determine the locus of the midpoint of the segment $[ PQ]$.

[i]Gheorghe Szolosy[/i]
1. **Identify the orthocenter and altitudes:**
   Let \( H \) be the orthocenter of the acute triangle \( ABC \). The orthocenter is the point where the altitudes of the triangle intersect. Let \( BB' \) and \( CC' \) be the altitudes of the triangle, where \( B' \in AC \) and \( C' \in AB \).

2. **Define the variable line and intersection points:**
   A variable line \( \ell \) passing through \( H \) intersects the segments \( [BC'] \) and \( [CB'] \) at points \( M \) and \( N \), respectively.

3. **Construct perpendiculars and intersection points:**
   From points \( M \) and \( N \), draw perpendicular lines to \( \ell \). These perpendiculars intersect the altitudes \( BB' \) and \( CC' \) at points \( P \) and \( Q \), respectively.

4. **Determine the locus of the midpoint of \( [PQ] \):**
   We need to find the locus of the midpoint of the segment \( [PQ] \).

5. **Use properties of the orthocenter and altitudes:**
   Since \( H \) is the orthocenter, the altitudes \( BB' \) and \( CC' \) are concurrent at \( H \). The line \( \ell \) passing through \( H \) implies that \( M \) and \( N \) are reflections of each other with respect to \( H \).

6. **Analyze the perpendiculars from \( M \) and \( N \):**
   The perpendiculars from \( M \) and \( N \) to \( \ell \) will be symmetric with respect to \( H \). Therefore, the points \( P \) and \( Q \) will also be symmetric with respect to \( H \).

7. **Find the midpoint of \( [PQ] \):**
   Since \( P \) and \( Q \) are symmetric with respect to \( H \), the midpoint of \( [PQ] \) will lie on the line passing through \( H \) and will be equidistant from \( H \).

8. **Conclusion:**
   The locus of the midpoint of the segment \( [PQ] \) is the line passing through \( H \) and parallel to \( BC \). This is because the midpoint of \( [PQ] \) will always be on this line due to the symmetry of the construction.

The final answer is the line passing through \( \boxed{ H } \) and parallel to \( BC \).