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### 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
```
Associated Topics:
High School Conic Sections/Circles
High School Euclidean/Plane Geometry
High School Geometry

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