# Thread: Divison of planes by n lines [Proof by induction]

1. ## Divison of planes by n lines [Proof by induction]

Hi. Im a freshman student.

This question taken from the book by Matoussek and Nestril.

The statement goes:

Let us draw n lines in the plane in such a way that no two are parallel and no three intersect in a common point. Prove that the plane is divided into exactly n(n+1)/2 +1 parts by the lines
Can someone please hint me as to how to adress this problem. Answers would be appreciated too but use spoiler tags or different font color so that I may read it after a few attempts by myself.

The only conclusion I have drawn is that the addition of an extra line to a plane containing n lines splits (n+1) regions into two parts.

For example :

# One line on a plane:
Two parts

# Add the (n+1)-th line ( i.e, now there are two lines):
n+1 planes split (i.e, 1+1=2, 2 planes split to form total 4 planes)

#so on...

now what?

Similarly, consider n planes in the 3-dimensional space in general position (no two are parallel, any three have exactly one point in common, and no four have a common point). What is the number of regions into which these planes partition the space?
Havnt really reached this part.

2. u r almost done
for the basis, it is 1 region for 0 lines
for inductive step,
every k th line adds k regions to the sum.
thus we have 1+ sigma(k) regions for k lines

3. Originally Posted by integerfan
u r almost done
for the basis, it is 1 region for 0 lines
for inductive step,
every k th line adds k regions to the sum.
thus we have 1+ sigma(k) regions for k lines