Question 6
Let $ABC$ be an acute scalene triangle inscribed in $(O)$, with orthocenter $H$. Let $M,N,P$ be the midpoints of $BC,CA,AB$, and let $D,E,F$ be the feet of the altitudes from $A,B,C$, respectively. Let $K$ be the reflection of $H$ in $BC$. Lines $DE,MP$ meet at $X$, and lines $DF,MN$ meet at $Y$.
(a) Line $XY$ meets the minor arc $BC$ of $(O)$ at $Z$. Prove that $K,Z,E,F$ are concyclic.
(b) Lines $KE,KF$ meet $(O)$ again at $S,T$, respectively, with $S,T\ne K$. Prove that $BS,CT,XY$ are concurrent.
(a) Line $XY$ meets the minor arc $BC$ of $(O)$ at $Z$. Prove that $K,Z,E,F$ are concyclic.
(b) Lines $KE,KF$ meet $(O)$ again at $S,T$, respectively, with $S,T\ne K$. Prove that $BS,CT,XY$ are concurrent.