On each segment, there are no more than two points. On the other hand, each intersection point belongs to at least two segments. Therefore, there are no more than $10 \cdot 2: 10=10$ points. An example with 10 points is shown in the figure.
![](https://cdn.mathpix.com/cropped/2024_05_06_9bfea73962495faf8ed9g-25.jpg?height=370&width=1352&top_left_y=1278&top_left_x=364)

Answer

10 points.
[ Systems of points and segments. Examples and counterexamples ] [ Classical combinatorics (miscellaneous). ]

On a plane, 10 equal segments were drawn, and all their points of intersection were marked. It turned out that each point of intersection divides any segment passing through it in the ratio $3: 4$. What is the maximum possible number of marked points?