Question 97
Let $ABC$ be a triangle and let the points $D \in (BC)$, $E \in (AC)$, $F \in (AB)$ be such that
$$
\frac{DB}{DC} = \frac{EC}{EA} = \frac{FA}{FB}.
$$
The halflines $(AD, BE$, and $(CF$ intersect the circumcircle of $ABC$ at points $M, N$ and $P$. Prove that the triangles $ABC$ and $MNP$ share the same centroid if and only if the areas of the triangles $BMC, CNA$ and $APB$ are equal.
$$
\frac{DB}{DC} = \frac{EC}{EA} = \frac{FA}{FB}.
$$
The halflines $(AD, BE$, and $(CF$ intersect the circumcircle of $ABC$ at points $M, N$ and $P$. Prove that the triangles $ABC$ and $MNP$ share the same centroid if and only if the areas of the triangles $BMC, CNA$ and $APB$ are equal.