Question 4
Let $ABC$ be an acute scalene triangle with altitudes $AD,BE,CF$, where $D\in BC,E\in CA,F\in AB$. Let $H,O,I$ be its orthocenter, circumcenter and incenter. Let $M,N,P$ be the midpoints of $BC,CA,AB$, and put $X=AI\cap NP$, $Y=BI\cap PM$, $Z=CI\cap MN$.
a) Prove that the circumcircles of $AXD,BYE,CZF$ have two common points lying on $OH$.
b) Lines $XP,YM,ZN$ meet these three circles again at $X',Y',Z'$, respectively, distinct from $X,Y,Z$. Let $J$ be the reflection of $I$ in $O$. Prove that $X',Y',Z'$ are collinear on a line perpendicular to $HJ$.
a) Prove that the circumcircles of $AXD,BYE,CZF$ have two common points lying on $OH$.
b) Lines $XP,YM,ZN$ meet these three circles again at $X',Y',Z'$, respectively, distinct from $X,Y,Z$. Let $J$ be the reflection of $I$ in $O$. Prove that $X',Y',Z'$ are collinear on a line perpendicular to $HJ$.