42.1. In parallelogram $A B C D$, the midpoint $K$ of side $A B$ is connected to vertex $C$, and the midpoint $M$ of side $B C$ is connected to vertex $D$. In what ratio does segment $K C$ divide segment $M D$?

$(8-10$ grades $)$
42.1. Let $E$ be the point of intersection of segments $K C$ and $M D$ (Fig. 9).

![](https://cdn.mathpix.com/cropped/2024_05_21_f0d089db3d527e92a992g-24.jpg?height=268&width=512&top_left_y=460&top_left_x=772)

Fig. 9

Draw a line through point $M$ parallel to $A B$ and denote the point of intersection of this line with segment $K C$ as $P$. From the similarity of triangles $P E M$ and $C E D$, we have: $M E: E D = M P: C D$. But $M P = 0.5 \cdot B K = 0.25 \cdot C D$, i.e., $M P: C D = 1: 4$. Therefore, $M E: E D = 1: 4$.