## 12. Figure

On a plane, there are five points such that no three of them lie on the same line. The points are connected by straight lines. What is the number of intersection points of the lines (excluding the original five points), if none of the drawn lines are parallel to each other?
12. Let $A, B, C, D$ and $E$ be five given points. The line $A B$ is intersected by the lines forming three sides of the triangle $C D E$. This gives three points of intersection. The same applies to each other line. The total number of lines is equal to the number of ways to choose two points from five, i.e., 10. Thus, we get $3 \cdot 10=30$ points, but each point is counted twice (why?). In fact, the actual number of intersection points of the lines is only 15, not 30.

## 154

![](https://cdn.mathpix.com/cropped/2024_05_21_fe999c0fe2ad81fc5164g-156.jpg?height=985&width=1077&top_left_y=250&top_left_x=501)