Problem 1591

1. Identify Key Points and Definitions:
- Let $H$ be the orthocenter of $\triangle ABC$.
- $M$ is the midpoint of the minor arc $BC$ on the circumcircle $(ABC)$.
- $M_1$ is the reflection of $M$ with respect to side $BC$.

2. Symmetry and Reflection:
- Reflecting $M$ across $BC$ to get $M_1$ implies that $M_1$ lies on the circumcircle $(BHC)$ because reflecting $M$ across $BC$ maps the circumcircle $(ABC)$ to $(BHC)$.

3. Homothety Argument:
- Consider a homothety centered at $A$ with a coefficient of $\frac{1}{2}$. This homothety maps the circumcircle $(BHC)$ to the nine-point circle of $\triangle ABC$.
- Under this homothety, the point $M_1$ maps to the midpoint of $AM_1$.

4. Nine-Point Circle:
- The nine-point circle of $\triangle ABC$ passes through the midpoints of the sides of $\triangle ABC$, the feet of the altitudes, and the midpoints of the segments joining the orthocenter $H$ to the vertices of $\triangle ABC$.
- Since the homothety maps $M_1$ to the midpoint of $AM_1$, and the nine-point circle is invariant under this homothety, the midpoint of $AM_1$ must lie on the nine-point circle.

5. Conclusion:
- Therefore, the nine-point circle bisects $AM_1$.

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