Question 172
Given an acute, scalene triangle $ABC$ with circumcircle $(O)$ and orthocenter $H$. Let $M$, $N$ and $P$ be the midpoints of $BC$, $CA$ and $AB$ and $D$, $E$ and $F$ be the feet of the altitudes from $A$, $B$ and $C$ of triangle $ABC$. Let $K$ be the reflection of $H$ through $BC$. Two lines $DE$, $MP$ intersect at $X$ and two lines $DF$, $MN$ intersect at $Y$.
a) Line $XY$ intersects the minor arc $BC$ of $(O)$ at $Z$. Prove that $K$, $Z$, $E$ and $F$ are concyclic.
b) Lines $KE$, $KF$ meet $(O)$ at $S$, $T$. Prove that $BS$, $CT$ and $XY$ are concurrent.
a) Line $XY$ intersects the minor arc $BC$ of $(O)$ at $Z$. Prove that $K$, $Z$, $E$ and $F$ are concyclic.
b) Lines $KE$, $KF$ meet $(O)$ at $S$, $T$. Prove that $BS$, $CT$ and $XY$ are concurrent.