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 2 of 5: Chords in one hexagon

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In plain words

Central symmetry pairs ordinary chords; regular-hexagon geometry handles diameters.

A,B\in H_i\Rightarrow AB\parallel CD\text{ for some distinct }C,D\in H_i

Detailed analysis

Every chord of a regular hexagon has a distinct parallel chord: central reflection handles every non-diameter chord, while each of the three diameters is parallel to a pair of opposite sides.