2. (10 points) In triangle $A B C$, a point $D$ is marked on side $B C$, and a point $M$ is marked on segment $A D$ such that triangles $A M B$ and $A M C$ have equal area. Which of the following statements are true?

a) Point $D$ must be the midpoint of side $B C$.

б) Point $D$ may not be the midpoint of side $B C$.

в) Point $M$ must be the midpoint of segment $A D$.

2) Point $M$ may not be the midpoint of segment $A D$.
Answer: a), g)

Solution: Rewrite the condition of the problem as

$$
1=\frac{S_{\triangle A M B}}{S_{\triangle A M C}}=\frac{\frac{1}{2} A B \cdot A M \cdot \sin \angle B A M}{\frac{1}{2} A C \cdot A M \cdot \sin \angle C A M}=\frac{A B \cdot \sin \angle B A M}{A C \cdot \sin \angle C A M}
$$

The right-hand side of this equality does not depend on the position of point $M$ on segment $A D$. From this, two conclusions follow. Firstly, statement g) is true, and c) is false. Secondly, the condition of the problem will still be satisfied if we take $M=D$. The triangles $A B D$ and $A C D$ have a common height, and their bases are $B D$ and $C D$, respectively. Therefore, due to the equality of the areas of the triangles, $B D=C D$. Thus, statement a) is true, and b) is false.