MHB Are All Points Collinear in This Week's University POTW?

[Ackbach]

Here is this week's problem:

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You are given a finite number of points in space with the property that any line that contains two of these points contains three of them. What must be true of all the points? Prove it.

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[Ackbach]

No one answered this week's University POTW. Here is my solution:

Spoiler

All the points must be collinear. We proceed by contradiction. (We assume there are more than some minimum number of points.) Suppose there are two points, $A$ and $B$ on a line. Because the number of points is finite, we will suppose that there is one point, $D$, not on the line, whose distance to the line is a minimum. Now the line $AB$, by hypothesis, must have a third point on it, call it $C$. Consider the distance from $D$ to $AB$, and call it $d_{\min}$. Now consider the "point" Q on $AB$ such that $QD=d_{\min}$. I use quotes, because $Q$ may not be a point in the geometry. At least two of $A,B,C$ must lie on the same side of $Q$, or possibly one of them coincides with it. WLOG, we assume $A$ and $C$ lie on the same side of the line with respect to $Q$. Draw the line $DA$. Then the distance from $C$ to $DA$ must be strictly less than $d_{\min}$, because it is less than or equal to the distance from $Q$ to $DA$, which is strictly less than $d_{\min}$. This contradicts the minimality of $d_{\min}$, hence there is no such minimum distance, and all the points must be collinear.