Question 56
Let $M$, $N$ and $P$ be points on the sides $AB$, $BC$ and $CA$ of $\triangle ABC$, respectively. The lines through $M$, $N$ and $P$, parallel to $BC$, $AC$ and $AB$, respectively, meet at a point $T$. Prove that:
a) if $\frac{AM}{MB} = \frac{BN}{NC} = \frac{CP}{PA}$, then $T$ is the centroid of $\triangle ABC$;
b) $S_{MNP} \leq \frac{1}{3} S_{ABC}$.
a) if $\frac{AM}{MB} = \frac{BN}{NC} = \frac{CP}{PA}$, then $T$ is the centroid of $\triangle ABC$;
b) $S_{MNP} \leq \frac{1}{3} S_{ABC}$.