# How many lines are determined by 8 points, no more than 3 of which are collinear? How many triangles are determined by the same points if no more than 4 are coplanar?

##### 1 Answer

I count 14 lines

#### Explanation:

**Collinear points**

We have 8 points (and just to have them named, we'll have

A line is defined by 2 points, and so we can have lines

Let's consider the case where we have

Let's go one further. Let's take line

So what lines have we made so far that have 3 points collinear?

Following this pattern, what other lines with 3 collinear points can we make? As I'm working through this, I'm keeping in mind that the maximum number of times I'll be able to list each point is three:

I don't think I can make any more lines like this, so we have 7 with three collinear points.

And now let's make the remaining lines. These will be two point lines that haven't been connected by our three point lines:

And so I count 14 lines in total.