Question 106
Let $ABC$ be a scalene triangle, $H$ be its orthocenter and $G$ be its centroid. Let $A_b$ and $A_c$ be points on $AB$ and $AC$, respectively, such that $B$, $C$, $A_b$, $A_c$ are cyclic and the points $A_b$, $A_c$, $H$ are collinear. Let $O_a$ be the circumcenter of the triangle $AA_bA_c$. Define $O_b$ and $O_c$ analogously. Prove that the centroid of the triangle $O_aO_bO_c$ lies on the line $HG$.