Question 40
Let $M$ be the centroid of a non-equilateral triangle $ABC$. Let $A$ and $A'$ lie on opposite sides of the line $BC$ such that the triangle $BCA'$ is equilateral, and let $A''$ be such an internal point of the segment $AA'$ that $A''A' = 2AA''$. Let the points $B', B'', C', C''$ be defined analogously. Prove that the triangle $A''B''C''$ is equilateral centred at $M$.