Question 49
$\mathrm{ABC}$ is an acute-angled triangle with orthocenter $\mathrm{H}$. $\mathrm{AE}$ and $\mathrm{BF}$ are altitudes. $\mathrm{AE}$ is reflected in the angle bisector of angle $\mathrm{A}$ and $\mathrm{BF}$ is reflected in the angle bisector of angle $\mathrm{B}$. The two reflections intersect at $\mathrm{O}$. The rays $\mathrm{AE}$ and $\mathrm{AO}$ meet the circumcircle of $\mathrm{ABC}$ at $\mathrm{M}$ and $\mathrm{N}$ respectively. $\mathrm{P}$ is the intersection of $\mathrm{BC}$ and $\mathrm{HN}$, $\mathrm{R}$ is the intersection of $\mathrm{BC}$ and $\mathrm{OM}$, and $\mathrm{S}$ is the intersection of $\mathrm{HR}$ and $\mathrm{OP}$. Show that AHSO is a parallelogram.