A generalization of a Parallelogram Theorem to a Parallelo-hexagon Inequality

(Relations between the sides and major diagonals of parallelo-hexagons)

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 is a parallelo-hexagon ABCDEF. Drag any of the red vertices and consider the relationships between the following displayed calculations:
4(AB² + BC² + CD²)
AD² + BE² + CF²
3(AB² + BC² + CD²)
What conjectures can you make? Check if they hold when ABCDEF is concave or crossed. Can you explain WHY your conjectures are true? (I.e. prove them?)

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Parallelo-hexagon Inequality

Check your conjectures above by Clicking Here.

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Last update, Michael de Villiers, 9 April 2013.