1. Given a triangle $A B C$, let $M$ be the midpoint of side $B C$ and let $B ^ { \prime }$ and $C ^ { \prime }$ be two points, respectively, on side $A B$ and on side $A C$, such that $A B ^ { \prime } = \frac { 1 } { 3 } A B$ and $A C ^ { \prime } = \frac { 1 } { 3 } A C$. Prove that, if the segments $M B ^ { \prime }$ and $M C ^ { \prime }$ are congruent to each other, then so are the sides $A B$ and $A C$.