**Theorem**: In 3D space, if the opposite sides of a spatial 3D hexagon are parallel, then the opposite sides are equal; i.e. the formed spatial figure is a 3D *parallelo-hexagon*.

**Corollary**: Unlike the plane, where it's possible to construct a hexagon with opposite sides parallel, but not necessarily equal, the above theorem proves the surprising result that this is impossible for a hexagon in 3D space.

**Illustration**: Above is a short video clip that briefly & dynamically illustrates the 3D parallelo-hexagon using Cabri 3D.

One can think of the 3D parallelo-hexagon as consisting of 3 different planar 2D parallelograms in space, all three intersecting in a single point; i.e. its point of symmetry.

**More Info**

a) For more background & proofs, read the paper by Heinz Schumann (University of Weingarten, Germany) & myself in the July 2018 issue of '*The Mathematical Gazette*', which is available at: *A surprising 3D result involving a hexagon*.

b) Below is a photo of a folded paper model of a 3D-parallelo-hexagon.

**Notes**:

1) The 3D parallelo-hexagon has a point of (reflective) symmetry like the 2D parallelogram, but unlike the 2D parallelogram, it does NOT have half-turn symmetry.

2) The converse of the theorem above is false. Having opposite sides equal for a spatial 3D hexagon is a necessary, but not sufficient condition. One can easily construct a counter-example as shown below from the original 3D parallelo-hexagon. By reflecting A_{1}A_{2}A_{6} in the plane on which parallelogram A_{2}A_{3}A_{5}A_{6} lies, a new spatial 3D hexagon A'_{1}A_{2}A_{3}A_{4}A_{5}A_{6} is obtained, which clearly still has the pairs of opposite sides equal, but A'_{1}A_{2} is no longer parallel to opposite side A_{4}A_{5}, and inclined at angle. Similarly, A'_{1}A_{6} is also no longer parallel to opposite side A_{3}A_{4}.

A 3D parallelo-hexagon is also formed by the centroids of the triangles subdividing a spatial 3D hexagon. See for example: *A generalization of Varignon's Theorem to 2n-gons*.

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Michael de Villiers, created 6 January 2019; updated 17 August 2021.