Question 55
Let $M$ be an interior point of $\triangle ABC$. The lines $AM$, $BM$ and $CM$ meet the lines $BC$, $CA$ and $AB$ at points $A_1$, $B_1$ and $C_1$, respectively, such that $S_{CB_1M} = 2 S_{AC_1M}$. Prove that $A_1$ is the midpoint of the segment $BC$ if and only if $S_{BA_1M} = 3 S_{AC_1M}$.