Question 3
Let $ABC$ be an acute scalene triangle inscribed in $(O)$, with orthocenter $H$. Let $E,F$ be the feet of the altitudes from $B,C$, respectively, and let $AH$ meet $(O)$ again at $D\ne A$.
a) Let $I$ be the midpoint of $AH$, and set $M=EI\cap BD$, $N=FI\cap CD$. Prove $MN\perp OH$.
b) Lines $DE,DF$ meet $(O)$ again at $P,Q\ne D$, respectively. The circumcircle of $AEF$ meets $(O)$ and $AO$ again at $R,S\ne A$, respectively. Prove that $BP,CQ,RS$ are concurrent.
a) Let $I$ be the midpoint of $AH$, and set $M=EI\cap BD$, $N=FI\cap CD$. Prove $MN\perp OH$.
b) Lines $DE,DF$ meet $(O)$ again at $P,Q\ne D$, respectively. The circumcircle of $AEF$ meets $(O)$ and $AO$ again at $R,S\ne A$, respectively. Prove that $BP,CQ,RS$ are concurrent.