5. In a convex quadrilateral $A B C D$, the areas of triangles $A B D$ and $B C D$ are equal, and the area of $A C D$ is half the area of $A B D$. Find the length of the segment $C M$, where $M$ is the midpoint of side $A B$, if it is known that $A D=12$.
Answer: $C M=18$. Solution. From the equality of the areas of $A B C$ and $C B D$, it follows that $O$ - the intersection point of the diagonals $A C$ and $B D$ bisects $A C$ (see figure). And from the fact that $S(A C D) = \frac{1}{2} S(A B D)$, it follows that $S(A O D) = \frac{1}{3} S(A O B)$, therefore, $O D = \frac{1}{3} B O = G O$, where $G$ is the point of intersection of the medians of triangle $A B C$.

Thus, $A G C D$ is a parallelogram, so $A D = C G = \frac{2}{3} C M = 12$, from which $C M = 18$.

![](https://cdn.mathpix.com/cropped/2024_05_06_df8875f405fedc435c59g-3.jpg?height=849&width=786&top_left_y=161&top_left_x=493)