Question 7
Let $ABC$ be an acute scalene triangle with centroid $G$, inscribed in a circle $(O)$. Let $H_a,H_b,H_c$ be the feet of the altitudes from $A,B,C$, and let $D,E,F$ be the midpoints of $BC,CA,AB$, respectively. The rays $GH_a,GH_b,GH_c$ meet $(O)$ at $X,Y,Z$, respectively.
(a) Prove that the circumcircle of $XCE$ passes through the midpoint of $BH_a$.
(b) Let $M,N,P$ be the midpoints of $AX,BY,CZ$, respectively. Prove that $DM,EN,FP$ are concurrent.
(a) Prove that the circumcircle of $XCE$ passes through the midpoint of $BH_a$.
(b) Let $M,N,P$ be the midpoints of $AX,BY,CZ$, respectively. Prove that $DM,EN,FP$ are concurrent.