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.
Unlike the plane, where it's possible to construct a hexagon with opposite sides parallel, but not necessarily equal (see for example, Exploration 1 at Easy Hexagon Explorations), the above theorem proves the surprising result that this is impossible for a hexagon in 3D space. In 3D if a (spatial) hexagon has opposite sides parallel, then it is a (spatial) parallelo-hexagon.
Illustration: Above is a short video clip that briefly & dynamically illustrates the 3D parallelo-hexagon using Cabri 3D.
Notes
1) 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.
2) Below is a photo of a folded paper model of a 3D-parallelo-hexagon.
3) 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.
4) 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 A1A2A6 in the plane on which parallelogram A2A3A5A6 lies, a new spatial 3D hexagon A'1A2A3A4A5A6 is obtained, which clearly still has the pairs of opposite sides equal, but A'1A2 is no longer parallel to opposite side A4A5, and inclined at angle. Similarly, A'1A6 is also no longer parallel to opposite side A3A4.
5) 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.
Reference
De Villiers, M. & Heinz Schumann, H. (2018). A surprising 3D result involving a hexagon. The Mathematical Gazette, Vol. 102, No. 554 (July), pp. 328-330.
Related Links
Easy Hexagon Explorations
Some Parallelo-hexagon Area Ratios
Parallelo-hexagon with Obtuse Angles
Triangle Centroids of a Hexagon form a Parallelo-Hexagon: A generalization of Varignon's Theorem
A generalization of a Parallelogram Theorem to Parallelo-hexagons, Hexagons and 2n-gons in general
2D Generalizations of Viviani's Theorem
Circumscribed Hexagon Alternate Sides Theorem
Cyclic Hexagon Alternate Angles Sum Theorem
Semi-regular Angle-gons and Side-gons: Generalizations of rectangles and rhombi
Alternate sides cyclic-2n-gons and Alternate angles circum-2n-gons
Pompe's Hexagon Theorem
Area ratios of some polygons inscribed in quadrilaterals and triangles
Jha and Savarn’s hexagon generalisation of Napoleon’s theorem
Dao Than Oai’s hexagon generalization of Napoleon’s theorem
A 1999 British Mathematics Olympiad Problem and its dual
Parallel-Hexagon Concurrency Theorem
Toshio Seimiya Theorem: A Hexagon Concurrency result
A theorem involving the perpendicular bisectors of a hexagon with opposite sides parallel
Haag Hexagon and its generalization to a Haag Polygon
Haag Hexagon - Extra Properties
Concurrency, collinearity and other properties of a particular hexagon
Conway’s Circle Theorem as special case of Side Divider Theorem
Fermat-Torricelli Point Generalization (aka Jacobi's theorem) plus Hexagon Generalization
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Michael de Villiers, created 6 January 2019; updated 17 August 2021; 6 Dec 2025.