3. (10 points) The diagonals $A C$ and $B D$ of quadrilateral $A B C D$ intersect at point $P$, and a point $M$ is marked on diagonal $A C$ such that $A M = M C$. Mark the correct statements.

a) The distances from point $M$ to the ends of diagonal $B D$ are equal: $B M = M D$.

b) The sum of the areas of triangles $A B M$ and $A M D$ is equal to the sum of the areas of triangles $B M C$ and $C M D$.

c) The sum of the areas of triangles $A B M$ and $C P D$ is equal to the sum of the areas of triangles $B M C$ and $A P D$.

d) None of the listed statements are correct.
Answer: b).
![](https://cdn.mathpix.com/cropped/2024_05_06_f23fd9be54420f8c770fg-2.jpg?height=500&width=954&top_left_y=1482&top_left_x=584)

Solution: Let's consider each statement.

a) The equality $B M = D M$ is equivalent to the point $M$ lying on the perpendicular bisector of $B D$. The example on the left shows a quadrilateral $A B C D$ for which this is not the case. Therefore, statement a) is incorrect.

b) The triangles $A B M$ and $B M C$ have equal bases $A M$ and $M C$, as well as equal heights dropped from vertex $B$. Therefore, the areas of these triangles are equal. Similarly, the areas of triangles $A M D$ and $C M D$ are equal. Thus,

$$
S_{\triangle A B M} + S_{\triangle A M D} = S_{\triangle B M C} + S_{\triangle C M D}
$$

which means b) is correct.
c) By b), $S_{\triangle A B M} = S_{\triangle B M C}$. Therefore, c) is equivalent to the equality of the areas of $\triangle C P D$ and $\triangle A P D$. The heights of these triangles are equal to the perpendicular dropped from vertex $D$ to the line $A C$. However, the bases $C P$ and $A P$ of the triangles do not necessarily coincide (as shown in the example on the right). Therefore, statement c) is incorrect.