Question 3
Let $ABC$ be an acute scalene triangle inscribed in $(O)$, with orthocenter $H$. Line $AH$ meets $(O)$ again at $D\ne A$. Let $E,F$ be the midpoints of $AB,AC$, respectively. The line through $H$ perpendicular to $HF$ meets $BC$ at $K$.
a) Line $DK$ meets $(O)$ again at $Y\ne D$. Prove that the intersection of $BY$ with the perpendicular bisector of $BK$ lies on the circumcircle of $OFY$.
b) The line through $H$ perpendicular to $HE$ meets $BC$ at $L$. Line $DL$ meets $(O)$ again at $Z\ne D$. Let $M,N,P$ be the intersections of the pairs $(BZ,OE)$, $(CY,OF)$, $(BY,CZ)$, respectively. Put $T=YZ\cap MN$, and let $d$ be the line through $T$ perpendicular to $OA$. Prove that $d$ passes through the midpoint of $AP$.
a) Line $DK$ meets $(O)$ again at $Y\ne D$. Prove that the intersection of $BY$ with the perpendicular bisector of $BK$ lies on the circumcircle of $OFY$.
b) The line through $H$ perpendicular to $HE$ meets $BC$ at $L$. Line $DL$ meets $(O)$ again at $Z\ne D$. Let $M,N,P$ be the intersections of the pairs $(BZ,OE)$, $(CY,OF)$, $(BY,CZ)$, respectively. Put $T=YZ\cap MN$, and let $d$ be the line through $T$ perpendicular to $OA$. Prove that $d$ passes through the midpoint of $AP$.