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- Thread starter Vishalrox
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- #2

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Do you mean the maximum number of points where they all intersect? Or just at least 2 intersect?

- #3

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6 intersects + 4 intersects + 2 intersects= 12

A straight line can intersect a circle a maximum of twice and there are four circles so:

8 intersects per line x 4 lines = 32 intersects.

Each line can also intersect the other lines at a single point so they overlap each of the four lines so not recounting an intersect would be:

3+2+1=6

Add them up:

12+32+6= 50 max intersects

Someone should check this ;)

- #4

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That's certainly an upper bound, but it's not immediately obvious that all these intersections are achievable simultaneously.A circle can intersect another circle only twice so you would have:

6 intersects + 4 intersects + 2 intersects= 12

A straight line can intersect a circle a maximum of twice and there are four circles so:

8 intersects per line x 4 lines = 32 intersects.

Each line can also intersect the other lines at a single point so they overlap each of the four lines so not recounting an intersect would be:

3+2+1=6

Add them up:

12+32+6= 50 max intersects

Start with some circle C. (This is not one of THE circles, it's just used for construction.)

You can arrange N equal circles, larger than C, such that each surrounds C. Clearly each pair of these intersects.

Any M straight lines in general position must intersect each other.

The region in which the intersections of the lines occurs can be bounded by a circle, D. Shrink that picture as necessary and fit D inside C. Necessarily, every line intersects every circle.

So any number of circles and lines can be arranged to achieve the upper bound.

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