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gsingh2011

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In summary: This is a complicated question. It would help if you could give more information about what you are trying to do.In summary, it seems that if you can find a new chord that intersects every other chord in a circle, then you can create new regions.

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gsingh2011

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quasar987

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gsingh2011

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quasar987 said:

Ok, so the r+1 chord creates r+1 new regions? How do I show that it is possible to draw a chord that intersects the previous r chords?

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csco

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quasar987

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gsingh2011 said:Ok, so the r+1 chord creates r+1 new regions? How do I show that it is possible to draw a chord that intersects the previous r chords?

This is proved by induction. It is obvious for r=1. Now assume it is true for r-1, and consider a circle with r chords each intersecting one another.

First rotate the circle about its center until none of the r chords are horizontal. Then,

Complete the proof.

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gsingh2011

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quasar987 said:This is proved by induction. It is obvious for r=1. Now assume it is true for r-1, and consider a circle with r chords each intersecting one another.

First rotate the circle about its center until none of the r chords are horizontal. Then,sweepthe circle with a horizontal chord. What will happen? It will eventually encounter a chord, say C1. Continue sweeping. It will encounter a second chord, C2. And by the time this happens, the horizontal chordstillintersects chord C1, otherwise it means we have swept chord C1 entirely without encountering chord C2! This is in contradiction with our induction hypothesis that every chord intersects each other ones.

Complete the proof.

By sweep you mean draw right? And I understand what you're saying, but why would having this new chord not intersect C1 and C2 lead to a contradiction?

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quasar987

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If by the time the new chord C intersects C2 for the first time, C no longer intersect C1, then it means that C has swept over the whole of C1 without intersecting C2. But C1 is supposed to intersect C2, so C should have intersected C1 and C2 simultaneously at some point.

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gsingh2011

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Ok, thanks, I think I should be able to figure it out now.

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quasar987

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Good luck!

A circle where every chord intersects every other chord is a special type of circle known as a "complete graph." In this type of circle, every point is connected to every other point by a chord, creating a continuous pattern.

A circle where every chord intersects every other chord has n(n-1)/2 chords, where n is the number of points or vertices on the circle. For example, a circle with 6 points would have (6*5)/2 = 15 chords.

A circle where every chord intersects every other chord has been studied in mathematics and has various applications in graph theory, computer science, and network analysis. It is also a visually interesting and symmetrical shape.

While it is not possible to create a perfect circle with infinitely many points in real life, this concept can be approximated with a large number of points. It has been observed in nature, such as in the structure of some flowers and the arrangement of atoms in certain crystals.

Yes, there are other shapes that exhibit this property, such as a regular polygon with an odd number of sides. However, a circle is the only shape where every chord is equal in length, making it unique in this regard.

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