Maximizing Intersection Points in a Circle: Geometry Math Problem Solution

[foxofdesert]

Homework Statement

then points are uniformly spaced on a circle. Each of the points is connected by a segment to exactly one of the other points for a total of five segments. Some pairs of the segments may intersect and some may not. What is the maximum possible number of distinct intersection points inside of the circle? Justify your answer.

Homework Equations

The Attempt at a Solution

I've been working on this question almost for a week, drawing a lot of circles, and trying.
I have found 8 intersections so far, but I keep thinking that there could be a 9th one.

Logically, there can be maximum of 10 points, since assuming there are 5 lines, the line 1 can have 4 intersections with other 4 lines, and the line 2 have 3, and so on. That is
n(n-1)/2 where n is the number of lines. However, since the end points of the line segements are uniformly spaced on the circle, these 10 intersection points are all in the same place, therefore 10 distinct intersections are impossible.

This is my idea so far, and haven't improved yet. Can anyone help me or give me some idea?
I will need an example of '9-intersection' and proof of 'there cannot be 10 intersections. I will keep posting if I get other thoughts.

 

[foxofdesert]

Here is the geometrical idea and some explain.

I drew a perpect pentagon first, then extended each line segments evenly, and drew a perpect circle. Then, I had thought that if I extend the line segeents even longer, then I could get 10 distinct intersect points. Then, however, the points cannot be uniformly placed on the circle, no matter how far I extend it.

 

[foxofdesert]

Is it that the problem is hard or my poor english skills? :(

Anyways, I concluded that there can be only 8 intersections, and my prof said it is correct. I proved that there cannot be 10 intersections when the points are uniformly placed, and did some weird-looking proof of there cannot be 9 intersections as well.

so, I got the answer, but I'm still curious about the full proof. anyone?

 

[checkitagain]

Homework Statement

$> > >$then points $< < <$are uniformly spaced on a circle.

Oh, I see you meant to type "ten points."




You did mention some difficulty with typing English.