Problem 2

Does there exist a finite set

M

of points in space, not all in one plane, such that for every two points

A,B\in M

there are two other points

C,D\in M

for which the lines

AB

and

CD

are parallel but distinct?

Step 3 of 5: Mixed pairs

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In plain words

Reflect both endpoints through the common center to reverse the displacement vector.

A\in H_1, B\in H_2, B\ne -A

Detailed analysis

The whole set is centrally symmetric. For a mixed pair that is not a diameter, choose $C=-A$ and $D=-B$ . Then $CD\parallel AB$ , and the line is distinct because the endpoints lie on the reflected, different chord.