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 $10 = 2n$ ?

#Combinatorics

Easy Math Editor

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$