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 4 of 5: Exceptional diameters

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In plain words

Only a diameter is unchanged as a line under central reflection, so isolate it.

B=-A

Detailed analysis

The only remaining pairs are the five diameters (the common one counted once). Each is already parallel to an opposite side of its own regular hexagon, by Step 2. Thus every pair has a distinct parallel partner.