If n points lie in a plane and no three are collinear, prove that there are 1/2n(n-1) lines joining these points. I'm not really sure where to even start with proving this. I'm supposed to use proof by induction. Any help would be great. Thanks.

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- Oct 11th 2013, 08:08 PMclintonh0610Proof by Induction
If n points lie in a plane and no three are collinear, prove that there are 1/2n(n-1) lines joining these points. I'm not really sure where to even start with proving this. I'm supposed to use proof by induction. Any help would be great. Thanks.

- Oct 11th 2013, 08:42 PMibduttRe: Proof by Induction
something appears to be wrong in the question because 1/2n(n+1) is a fraction and number of lines cannot be in fraction.

- Oct 12th 2013, 03:56 AMPlatoRe: Proof by Induction
There is nothing wrong with the formula. It just written poorly.

It is $\displaystyle \binom{n}{2}=\frac{n(n-1)}{2}$. To prove that by induction is a pain.

The base case $\displaystyle n=1$ is trivial: there is no line segment.

Suppose it is true for $\displaystyle n=K$, there are $\displaystyle K$ points and $\displaystyle \frac{K(K-1)}{2}$ line segments.

If you add one more point, then how many new line segments are added?

See if you can get $\displaystyle \frac{(K+1)(K)}{2}~?$