Five Noncollinear Points

Date: 2 Apr 1995 18:05:43 -0400
From: Daniel Gomez
Subject: Combinations/permutations

I am a ninth grader and am stuck on the following 
question from my math homework.  Can you help?

In general, how many 
   a.  undirected lines
   b.  circles
are determined by five noncollinear points?

Date: 2 Apr 1995 19:30:20 -0400
From: Dr. Ken
Subject: Re: Combinations/permutations

Hello there!

Well, the answer is that it depends on what you mean by 
"noncollinear," a word that different people use to mean 
different things (when talking about more than 3 points).  
If you mean that all five points don't lie on the same line, 
then you'll get one answer.  If you mean that no three of 
them lie on the same line, then that's another thing.  Here 
are a couple of configurations of five points that satisfy 
the first definition, but not the second:

_____________________________________________
1)                                o
         o       o          o           o           

_____________________________________________
2)
         o      o       o
                o
                o
_____________________________________________

Now here are five points that do satisfy the second 
definition:

_____________________________________________

                                 o

                             o       o

                             o       o
______________________________________________

Now, there's an axiom in geometry that any two points 
determine a unique line.  If no three points are collinear, 
that means that no two points will determine the same 
line as any other two points, so there will be no overlap.  
So the number of lines determined will be the number of
two-element subsets of this five-element set.

As for question B, there could again be several answers, 
no matter which definition you use for noncollinear.  For 
instance, draw a circle, and then pick five noncollinear 
points on it.  You can see that these five points all 
determine the same circle.  But if you play around with a 
compass, you'll also be able to convince yourself that you 
can make five points in the plane, each three of which 
determine a distinct circle.  Do you know how you could 
prove that?  It's a neat problem.

Anyway, I hope this helps you!

-Ken "Dr." Math