It’s easy to draw a line and scatter an equal number of points on either side. Is it always possible to do this in reverse order? That is, given a finite set of points in the plane, is it always possible to draw a line that divides it neatly in two? (If there are an odd number of points, assume the line must intercept precisely one of them.)
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Yes, it’s always possible. Join every pair of points with a line segment. Now, to the left of the field, draw a line that’s parallel to none of these segments. If we move the line rightward through the field, keeping its slope constant, then it will encounter one point at a time, so we can stop it precisely midway through the field (or anywhere else we like).
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