When you look at the six pairs of opposite sides, you may notice that the pairs themselves could be grouped into three sets of four that "look alike."
For example, the opposite sides formed by segment A12A14 and segment A23A34 seem to naturally go together with the opposite sides formed by segment A12A23 and segment A14A34.
Likewise, the opposite sides formed by segment A23A24 and segment A13A14 seem to naturally go together with the opposite sides formed by segment A13A23 and segment A14A24.
When you construct segments connecting the midpoints of these three sets of opposite sides, the segments intersect in collinear points.
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