Let $B$ and $C$ be two fixed points on a circle centered at $O$ that are not diametrically opposed. Let $A$ be a variable point on the circle distinct from $B$ and $C$ and not belonging to the perpendicular bisector of $BC$. Let $H$ be the orthocenter of $\triangle ABC$, and $M$ and $N$ be the midpoints of the segments $BC$ and $AH$, respectively. The line $AM$ intersects the circle again at $D$, and finally, $NM$ and $OD$ intersect at $P$. Determine the locus of points $P$ as $A$ moves around the circle.
1. **Identify the given elements and their properties:**
   - \( B \) and \( C \) are fixed points on a circle centered at \( O \).
   - \( A \) is a variable point on the circle, distinct from \( B \) and \( C \), and not on the perpendicular bisector of \( BC \).
   - \( H \) is the orthocenter of \(\triangle ABC\).
   - \( M \) is the midpoint of \( BC \).
   - \( N \) is the midpoint of \( AH \).
   - The line \( AM \) intersects the circle again at \( D \).
   - \( NM \) and \( OD \) intersect at \( P \).

2. **Determine the properties of the orthocenter \( H \):**
   - The orthocenter \( H \) of \(\triangle ABC\) is the point where the altitudes of the triangle intersect.
   - Since \( A, B, C \) lie on a circle, \( H \) lies inside the triangle.

3. **Analyze the line \( AM \) and its intersection with the circle:**
   - The line \( AM \) intersects the circle again at \( D \).
   - Since \( M \) is the midpoint of \( BC \), \( AM \) is a median of \(\triangle ABC\).

4. **Examine the properties of the midpoints \( M \) and \( N \):**
   - \( M \) is the midpoint of \( BC \).
   - \( N \) is the midpoint of \( AH \).

5. **Analyze the intersection of \( NM \) and \( OD \):**
   - \( NM \) and \( OD \) intersect at \( P \).

6. **Determine the locus of points \( P \):**
   - We need to show that the locus of \( P \) is an ellipse with foci \( M \) and \( O \).

7. **Use the properties of the ellipse:**
   - For an ellipse with foci \( M \) and \( O \), the sum of the distances from any point on the ellipse to the foci is constant.
   - We need to show that \( |MP| + |OP| = R \), where \( R \) is the radius of the circle.

8. **Analyze the angles and distances:**
   - Consider the angles \(\widehat{PMD}\), \(\widehat{NMA}\), and \(\widehat{OAM}\).
   - Since \( \widehat{PMD} \equiv \widehat{NMA} \equiv \widehat{OAM} \equiv \widehat{PDM} \), we have \( |PD| = |PM| \).

9. **Conclude the relationship between the distances:**
   - Since \( |PD| = |PM| \), we have \( |PM| + |OP| = |PD| + |OP| = |OD| = R \).

Therefore, the locus of points \( P \) as \( A \) moves around the circle is an ellipse with foci \( M \) and \( O \), and the sum of the distances from \( P \) to \( M \) and \( O \) is equal to the radius \( R \) of the circle.

\[
\boxed{|MP| + |OP| = R}
\]