(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. 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.

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# Geometry math problem

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