While searching through a textbook for interesting explorations for middle school students I came across the below chart. At first glance it seemed to be simply a fill in the blank no thinking exercise but as I investigated it a little further, a number of interesting patterns emerged.
 

Rules for creating and counting line segments.
In creating segments, do not cross another segment.For example, the red line is not to be used. If you can connect two points with a line segment, you must do so. In counting segments, do not double count overlapping segments. This picture represents two line segments, not three.

First Pattern - Minimum Segments

Obviously, when two points are provided the minimum number of segments that can be formed is one. When a new point is introduced, we have two choices as to where to place the point; on the line segment (or its extension) or off of the line. We find that by placing the point on the line segment or on its linear extension only one more line segment is formed, which is obviously still a minimum. This pattern continues on. Thus the process for creating a minimum number of segments is to create collinear points. A formula that you could use for calculating the minimum number of segments is minimum segments = (# of points) - 1
 

Second Pattern - Plane Division

Is it possible to represent all segment values between the minimum and maximum? This is a easier question to answer, then it is to prove. After a little investigation, I came to find that by dividing the plane into regions I was able to show which situations were possible. Thus enabling me to determine an answer to the above question.
 

As mentioned earlier two points create only one possible situation: one line segment. When we look for possible place for the third point we find that the plane gives us two possible locations.

(Left) Given two points we find two possible regions to place the third point. Region #1(Red) Anywhere off of the line or its extension. Region #2 (Black) The line or its extension. (Left Down) When we place the third point off of the line we notice the formation of two new segments giving us a total of three line segments for three points. (Down) When the third point is placed on the line or its extension we notice that only one new line segment is created, giving us a total of two line segments for three points.

Summary for three points - In this case the minimum is two while the maximum is three. Obviously we can represent all segment values between the minimum and the maximum because in this case there are no segment values between them.

3 POINTS
MINIMUM (2 segments)
MAXIMUM (3 segments)

When working with four points we have two possible arrangements to begin with. The case where 2 segments are formed by 3 points and the case where 3 segments are formed by 3 points.
CASE #1 -Working from the minimum arrangement

(Left) Given three points(minimum arrangement) we find two possible regions to place the third point. Region #1(Red) Anywhere off of the line or its extension. Region #2 (Black & Blue) The line or its extension. (Left Down) When we place the fourth point off of the line we notice the formation of three new segments giving us a total of five line segments for four points. (Down) When the fourth point is placed on the line or its extension we notice that only one new line segment is created, giving us a total of three line segments for four points.

Summary of Case #1 - Thus two possible values exist when we begin with the minimum arrangement for three points. One formation of 3 segments and another of 5 segments.

CASE #2 -Working from the maximum arrangement

(Right) Given the three points in their maximum arrangement(a triangle), we look for some insight on where to place our next point.... inside.... outside.....??

(Down) When working with this arrangement we find that two regions are created in the entire plane. Region #1 (Yellow) This region represents all points that are visible by the three vertices of the triangle. Three segments are formed from the vertices to this new point in region #1. Region #2 (Black & Green) This region has two sub-regions: the black region includes placing the new point on the sides of the triangle or their extension while the green region represents the shadowed regions where only two vertices are visible. Placing a point in these regions creates two new segments. Region #1 and Region #2 represent the entire plane.

Summary of Case #2 -Thus two possible values exist when we begin with the minimum arrangement for four points. One formation of 3 segments and another of 5 segments.

Summary for four points - In this case we saw a number of possible arrangements, 3 segments, 5 segments and 6 segments. Our original question was whether we could represent all possible segment values from the minimum to the maximum. Well here is a case where we can say that it is impossible to create a four segment arrangement with four points. We know it is impossible because we have represented every point in the plane and at no location can only four segments exist.

4 POINTS
MINIMUM (3 segments)
MIDDLE (5 segments)
MAXIMUM (6 segments)

Generalization for n points - To determine all possible formations follow the below steps:
1) Take the previous points arrangements (Minimum, Middle(s) and Maximum) and create a planar division display all possible point locations.
2) Determine the minimum, and the maximum segment values. The minimum is created by placing the new point on the line segment of the previous example minimum arrangement(see First Pattern - Minimum Segments). The maximum is created by placing the point in a three segment region of the previous examples maximum arrangement(see Third Pattern - Maximum Segments).
3) Determine the middle segment values. Create the regions for the previous examples middle arrangements. These will display all possible different combinations.

Third Pattern -Maximum Segments

Plane division helps us look at the pattern formed by the maximum segments. We see that we can form the maximum segments when we work off of the previous arrangements maximum formation. It is also a fairly direct step to see that the maximum is three more than the previous arrangements maximum. A formula that you could use for calculating the maximum segments = 3 * ((# of points) - 1)
 

Fourth Pattern - Number of Segments to the Number of Triangles Formed

A nice pattern exists but I will leave this to be explored by the reader.

Further Investigation: How would things be different if we could cross segments? How would our maximums and minimums be effected if we could count overlapping segments?...... Altering the original parameters would quite drastically change the above observations..... explore.. and find out more.....