Question 171
Given an acute, scalene triangle $ABC$ with circumcircle $(O)$, orthocenter $H$ and centroid $G$. Let $H_a$, $H_b$ and $H_c$ be the feet of altitudes from $A$, $B$ and $C$ in triangle $ABC$ and $D$, $E$ and $F$ be the midpoints of $BC$, $CA$ and $AB$ in that order. Let the rays $GH_a$, $GH_b$ and $GH_c$ intersect $(O)$ at $X$, $Y$ and $Z$ respectively.
a) Prove that the circumcircle of triangle $XCE$ passes through the midpoint of $BH_a$.
b) Let $M$, $N$ and $P$ be the midpoints of $AX$, $BY$ and $CZ$ respectively. Prove that $DM$, $EN$ and $FP$ are concurrent.
a) Prove that the circumcircle of triangle $XCE$ passes through the midpoint of $BH_a$.
b) Let $M$, $N$ and $P$ be the midpoints of $AX$, $BY$ and $CZ$ respectively. Prove that $DM$, $EN$ and $FP$ are concurrent.