Question 62
Let $ABC$ be an acute scalene 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 triangle $HEF$, with center $I$. Let $K,J$ be the midpoints of $BC,EF$, respectively. The line $HJ$ meets $(I)$ again at $G$, and $GK$ meets $(I)$ again at $L$.
Part (a). Prove that $AL$ is perpendicular to $EF$.
Part (b). Let $AL$ meet $EF$ at $M$, let $IM$ meet the circumcircle of triangle $IEF$ again at $N$, and let $DN$ meet $AB,AC$ at $P,Q$, respectively. Prove that $PE,QF,AK$ are concurrent.
Part (a). Prove that $AL$ is perpendicular to $EF$.
Part (b). Let $AL$ meet $EF$ at $M$, let $IM$ meet the circumcircle of triangle $IEF$ again at $N$, and let $DN$ meet $AB,AC$ at $P,Q$, respectively. Prove that $PE,QF,AK$ are concurrent.