Problem 1204

To solve this problem, we need to ensure that among any four points chosen from the set of 1001 points, we can always find a triangle. This means that we need to avoid the situation where all four points are connected in such a way that they form a quadrilateral without any diagonals.

1. Understanding the Problem:
- We have 1001 points in the plane, and no three of them are collinear.
- We need to draw segments such that among any four points, there is always a triangle.

2. Key Insight:
- If we connect every pair of points, we will certainly have a triangle among any four points because every set of four points will have six segments connecting them, which guarantees at least one triangle.

3. Calculating the Total Number of Segments:
- The total number of segments that can be drawn between 1001 points is given by the combination formula $\binom{1001}{2}$, which represents the number of ways to choose 2 points out of 1001 to form a segment.
- This is calculated as:
$$\binom{1001}{2} = \frac{1001 \cdot 1000}{2} = 500500$$

4. Ensuring the Minimum Number of Segments:
- To ensure that among any four points there is always a triangle, we need to consider the minimum number of segments required.
- If we do not connect every pair of points, we risk having a set of four points that form a quadrilateral without any diagonals, which would not satisfy the condition.
- Therefore, the minimum number of segments required is indeed the total number of segments possible, which is $500500$.

Conclusion:
$$\boxed{500500}$$