Question 170
Let $ABC$ be an acute, non-isosceles triangle with $(O)$ as its circumcircle. Denote $H$ as the orthocenter and $BE$, $CF$ as the altitudes of triangle $ABC$. Suppose that $AH$ intersects $(O)$ at $D$ different from $A$.
1.
Let $I$ be the midpoint of $AH$, $EI$ meets $BD$ at $M$ and $FI$ meets $CD$ at $N$. Prove that $MN$ is perpendicular to $OH$.
2.
The lines $DE$, $DF$ intersect $(O)$ at $P$, $Q$ respectively ($P$ and $Q$ are different from $D$). The circle $(AEF)$ intersects $(O)$ and $AO$ at $R$, $S$ respectively ($R$ and $S$ are different from $A$). Prove that $BP$, $CQ$, $RS$ are concurrent.
1.
Let $I$ be the midpoint of $AH$, $EI$ meets $BD$ at $M$ and $FI$ meets $CD$ at $N$. Prove that $MN$ is perpendicular to $OH$.
2.
The lines $DE$, $DF$ intersect $(O)$ at $P$, $Q$ respectively ($P$ and $Q$ are different from $D$). The circle $(AEF)$ intersects $(O)$ and $AO$ at $R$, $S$ respectively ($R$ and $S$ are different from $A$). Prove that $BP$, $CQ$, $RS$ are concurrent.