Question 118
A circle passing through the vertices $B$ and $C$ of the triangle $ABC$ meets again $AB$ and $AC$ at $C_1$ and $B_1$ respectively. Let $A'$ be the midpoint of $B_1C_1$ and $AA' \cap BC = \{A''\}$. Define similarly $B''$ and $C''$. Prove that the triangles $ABC$ and $A''B''C''$ have the same centroid.