An acute-angled non-isosceles triangle $ABC$ is drawn, a circumscribed circle and its center $O$ are drawn. The midpoint of side $AB$ is also marked. Using only a ruler (no divisions), construct the triangle's orthocenter by drawing no more than $6$ lines.

(Yu. Blinkov) 
1. **Identify Key Points and Lines:**
   - Let \( D \) be the midpoint of \( AB \).
   - Let \( O \) be the center of the circumscribed circle of \( \triangle ABC \).
   - Let \( H \) be the orthocenter of \( \triangle ABC \).

2. **Draw Line \( AO \):**
   - Draw the line \( AO \) and mark its intersection with the circumscribed circle at point \( A' \).

3. **Establish Parallel Lines:**
   - Note that \( \angle HBC = \angle ABO = \angle BAO = \angle BCA' \), which implies that \( HB \parallel CA' \).
   - By symmetry, \( HC \parallel A'B \).

4. **Form Parallelogram \( HCA'B \):**
   - Since \( HB \parallel CA' \) and \( HC \parallel A'B \), quadrilateral \( HCA'B \) is a parallelogram.
   - The center of this parallelogram is \( D \), the midpoint of \( AB \).

5. **Draw Line \( AD \):**
   - Draw the line \( AD \), which passes through \( H \).

6. **Draw Diameter \( DO \):**
   - Draw the diameter \( DO \) of the circumscribed circle. This line intersects the triangle \( \triangle ABC \) at point \( E \) on \( AC \).

7. **Reflect Point \( A \):**
   - Draw line \( BE \), which intersects the circle at point \( F \). Point \( F \) is the reflection of \( A \) about \( DO \).

8. **Draw Line \( FO \):**
   - Draw the line \( FO \) to intersect the circle at point \( G \).

9. **Find the Altitude:**
   - Line \( AG \) is an altitude of \( \triangle ABC \).

10. **Determine the Orthocenter:**
    - The orthocenter \( H \) of \( \triangle ABC \) is the intersection of lines \( AG \) and \( AD \).

The final answer is the intersection of lines \( \boxed{ AG } \) and \( AD \).