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If there are five points on a circle. How many line segments can be drawn on it, but without overlapping the regions?

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what is a region? – Carry on Smiling Aug 29 '12 at 21:23
Do you care to elaborate on the meaning of "regions"? You have connect the points in a pentagon like this, you can add more line segments, but they intersect like this. Finally, you can connect it all over the place like this. What do you have in mind? – user2468 Aug 29 '12 at 21:37
You may mean that the line segments join pairs of vertices (the $5$ given points), and that two line segments can only meet at a vertex. Then drawing a few picturs should convince one of the answer, even if in Middle School a proof is difficult to write down. – André Nicolas Aug 29 '12 at 22:07

The number of line segments you can draw, each segment joining two of the 5 points on a circle, no two segments intersecting except at the 5 points, is 7. The five line segments that join the 5 points in a convex pentagon can certainly be drawn, since they don't intersect any other line segments (except at the 5 points). Having drawn those 5 line segments, you can draw any 2 diagonals of the pentagon, but no more than 2.

If that's not the question you want answered, please consider editing your question to clarify.

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