Question 31
In triangle $A B C$, choose point $A_{1}$ on side $B C$, point $B_{1}$ on side $C A$, and point $C_{1}$ on side $A B$ in such a way that the three segments $A A_{1}$, $B B_{1}$, and $C C_{1}$ intersect in one point $P$. Prove that $P$ is the centroid of triangle $A B C$ if and only if $P$ is the centroid of triangle $A_{1} B_{1} C_{1}$.