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Dividing Line

This is inspired by Mehul Chaturvedi's question Equally divided 10 points.

Consider all sets of 10 points in the plane, in which no 3 are collinear. A line \(l\) that connects 2 of these points is a dividing line if it divides the remaining points into 2 equal regions of 4 points each. Over all configurations, what is the maximum number of dividing lines?

By a simple counting argument, we can show that there are at least 5 dividing lines in any configuration.

If the 10 points form a convex set (ie no points are in the interior of the convex hull), then we can prove that there are exactly 5 dividing lines.

To get more than 5, consider the vertices of a 9-gon, along with the center. Then, any vertex that is connected to the center gives us a dividing line, and so there are (at least) 9 such lines.

Can we do better?

Can we generalize this to \( 10 = 2n \)?

Note by Calvin Lin
2 years ago

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