Clarification & Extension of Points, Lines & Planes

***** Extension on the maximum number of intersections between lines *****
In today's class, we explored the maximum number of intersections n number of lines can have. We could see the pattern from a table, and were able to find out the maximum intersections for 7 and 10 lines. However, the pattern quickly got to be too cumbersome to use for values of n that are quite large. Some of you in the classes came up with a great idea --- We need a FORMULA expressed in terms of n. Fortunately, we did come up with one after some inductive reasoning, which is the process of observing patterns and making conjectures.

Here's a classic math question that is related with the above process: "How many different handshakes are possible in a room with 'n' people?" Can you use the same reasoning process to figure out a formula?

***** Clarification on Points, Lines and Planes *****

A Line extends in one dimension. Any given two points determine a line. In other words, between any two points exists a line.
A Plane extends in two dimensions. Any given three points determine a plane. In other words, there is always a plane for any given three points.

Let's think about the following statements. Pick which words to use.

• Three points must/might/will never be collinear.
• Three points must/might/will never be coplanar.
• Four points must/might/will never be collinear.
• Four points must/might/will never be coplanar.
  

It is sometimes difficult to visualize the relationship between points, lines and planes without an actual model. So let's look into this model of a cube.