Question 3
Let $ABC$ be an acute, non-isosceles triangle with orthocenter $H$, and let $D,E,F$ be the feet of the altitudes from $A,B,C$, respectively. Let $(I)$ be the circumcircle of $HEF$, with center $I$, and let $K,J$ be the midpoints of $BC,EF$. The line $HJ$ meets $(I)$ again at $G$, and $GK$ meets $(I)$ again at $L$.
a) Prove that $AL\perp EF$.
b) Let $M=AL\cap EF$. The line $IM$ meets the circumcircle of $IEF$ again at $N$, and $DN$ meets $AB,AC$ at $P,Q$, respectively. Prove that $PE,QF,AK$ are concurrent.
a) Prove that $AL\perp EF$.
b) Let $M=AL\cap EF$. The line $IM$ meets the circumcircle of $IEF$ again at $N$, and $DN$ meets $AB,AC$ at $P,Q$, respectively. Prove that $PE,QF,AK$ are concurrent.