Given seven points in the plane, some of them are connected by segments such that:

[b](i)[/b] among any three of the given points, two are connected by a segment;
[b](ii)[/b] the number of segments is minimal.

How many segments does a figure satisfying [b](i)[/b] and [b](ii)[/b] have? Give an example of such a figure.
1. **Understanding the Problem:**
   We are given seven points in the plane, and we need to connect some of them with segments such that:
   - Among any three points, at least two are connected by a segment.
   - The number of segments is minimal.

2. **Applying Turán's Theorem:**
   Turán's Theorem helps us find the maximum number of edges in a graph that does not contain a complete subgraph \( K_r \). For our problem, we need to ensure that among any three points, at least two are connected, which implies that the graph should not contain an independent set of three points.

3. **Graph Analysis:**
   Let \( G \) be a graph with 7 vertices (points) and \( \varepsilon \) edges (segments). We need to ensure that \( G \) satisfies the given conditions. According to Turán's Theorem, the maximum number of edges in a graph with 7 vertices that does not contain a \( K_3 \) (triangle) is given by:
   \[
   \varepsilon' \leq \left\lfloor \frac{7^2}{4} \right\rfloor = \left\lfloor \frac{49}{4} \right\rfloor = 12
   \]
   This means that the complement graph \( G' \) (where edges in \( G \) are non-edges in \( G' \)) can have at most 12 edges.

4. **Finding the Minimum Number of Segments:**
   The total number of possible edges in a complete graph \( K_7 \) with 7 vertices is:
   \[
   \binom{7}{2} = 21
   \]
   Since \( G' \) can have at most 12 edges, the minimum number of edges in \( G \) is:
   \[
   \varepsilon = 21 - 12 = 9
   \]

5. **Constructing the Minimal Configuration:**
   To achieve the minimal configuration, we can consider the union of a complete graph \( K_3 \) and a complete graph \( K_4 \). This configuration ensures that among any three points, at least two are connected, and it uses the minimal number of segments.

   - \( K_3 \) has 3 vertices and 3 edges.
   - \( K_4 \) has 4 vertices and 6 edges.
   - Together, they form a graph with 7 vertices and \( 3 + 6 = 9 \) edges.

Therefore, the minimal configuration satisfying the given conditions has 9 segments.

The final answer is \(\boxed{9}\)