A generalization of a Parallelogram Theorem to a Parallelo-hexagon and Hexagon Equality 3

(Relations between the sides and diagonals of parallelo-hexagons, and the general theorem of Douglas - 1981)

An interesting parallelogram theorem, apparently first noted and proved by Apollonius of Perga (ca. 262 BC - ca. 190 BC), states that the sum of the squares of the sides of a parallelogram is equal to the sum of the squares of its diagonals (see Parallelogram Law). It seems natural to consider its possible generalization to a hexagon with opposite sides equal and parallel, i.e. a parallelo-hexagon. Below are some interesting properties related to such an investigation.

1) In a hexagon ABCDEF with opposite sides equal and parallel, i.e. a parallelo-hexagon, and the diagonals are parallel to a pair of opposite sides, the following three relationships hold between its sides and diagonals:
AD² + BE² + CF² = 4(AB² + BC² + CD²)
3(AB² + BC² + CD²) = (AC² + CE² + EA²)
3(AD² + BE² + CF²) = 4(AC² + CE² + EA²)
Consider the accurately constructed dynamic sketch below. Drag any of the red vertices to explore.

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Fig. 1 Parallelo-hexagon with Diagonals parallel to Opposite sides - Three equalities

2) In a hexagon ABCDEF with opposite sides equal and parallel, i.e. a general parallelo-hexagon, with M and N the respective centroids of triangles ACE and BDF, the following three relationships hold between its sides, diagonals and the distance MN:
AD² + BE² + CF² ≤ 4(AB² + BC² + CD²)
AB² + BC² + CD² + AC² + CE² + EA² = AD² + BE² + CF²
4(AB² + BC² + CD²) - (AD² + BE² + CF²) = 9MN² (This result is a special case of the theorem of Douglas - see no. 3 below)
Consider the accurately constructed dynamic sketch below. Drag any of the red vertices to explore. Also check concave and crossed cases.

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Fig. 2 Parallelo-hexagon with Centroids - One inequality and two equalities

3) Remarkably, all the preceding results are merely special cases of Douglas' Theorem (1981) for 2n-gons. For a general hexagon ABCDEF with M and N the respective centroids of triangles ACE and BDF, the following relationship holds between its sides, diagonals and the distance MN:
AB² + BC² + CD² + DE² + EF² + FA² - AC² - BD² - CE² - DF² - EA² - FB² + AD² + BE² + CF² = 9MN²
Consider the accurately constructed dynamic sketch below. Drag any of the red vertices to explore. Also check concave and crossed cases.

Please install Java (version 1.4 or later) to use JavaSketchpad applets.

Fig 3 Douglas' theorem (1981) for sides and diagonals of 2n-gons

On online investigation for students is available in the section for Explorations for Students. Click Here to go there.

Read my paper in the July 2012 issue of the Mathematical Gazette at: Relations between the sides and diagonals of a set of hexagons.

Read Douglas' paper in the March 1981 issue of the Mathematical Gazette at: A generalization of Apollonius' theorem. Perhaps even more remarkable, is that the 2n points in Douglas' generalization do not have to be co-planar, but can be in 3D space. Also note that what he defines in his paper as 'orthocentres' is normally called 'centroids'; i.e. since they are obtained by the arithmetic average of the coordinates of the points.


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Created by Michael de Villiers, 8 April 2013.