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Suppose we have a finite number of points randomly placed on a plane. Is it always possible to rotate all the points such that every vertical line passes through at most one point?

I feel like this is true since there's only a finite number of points and an infinite number of possible rotations, but I am not quite convinced. If this is true how would we go about proving it?

If it's not true, can we add assumptions such as having no points be collinear to make this true?

Any help would be greatly appreciated.

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For each pair of points $P_i$ and $P_j$, consider the line $l_{i,j}$ joining them. There are only finitely many lines $l_{i,j}$, so there's a line $l'$ not parallel to any of them. Now rotate the plane so that $l'$ becomes vertical.

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Induction works easily.

Like you said, given any 2 points, there are infinitely many rotations that can be made, and finitely many that result in a vertical line through either of these 2 points, also passing through another in the set.

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