## Some Parallelo-hexagon Area Ratios

Theorem 1: If G, H, I, and J are the respective midpoints of the sides AB, BC, DE and EF of a parallelo-hexagon ABCDEF (a hexagon with opposite sides equal and parallel), then area ABCDEF = 2 area GHIJ.

Theorem 2: If the midpoints of all the sides of a parallelo-hexagon ABCDEF are connected as shown, then area ABCDEF = 4/3 area GHIJKL.
Click on the 'Link to Midpoints of all sides' button to navigate to the accompanying sketch.

Theorem 3: If G, H, and I, are the respective midpoints of the sides AB, CD, and EF of a parallelo-hexagon ABCDEF are connected as shown, then area ABCDEF = 8/3 area GHI.
Click on the 'Link to Midpoints of alternate sides' button to navigate to the accompanying sketch.

Explore
Investigate the theorems above with the dynamic sketches below. Use dragging to change their shapes & also to check whether the results hold when the parallelo-hexagons becomes concave or crossed.

#### .sketch_canvas { border: medium solid lightgray; display: inline-block; } Some Parallelo-hexagon Area Ratios

Challenge
Can you explain why (prove that) these area ratios remain invariant?

Further Generalization
Can you generalize these results further?
Consider, for example, a) generalizing to a parallelo-octagon, or b) what happens if say point G is not a midpoint, and GH, GJ and HI are drawn parallel to the same diagonals as before?
Compare your answers to b) with these generalizations by Nestor Sánchez León (Sept 2023) with dynamic sketches at: Generalización de razones sobre áreas de hexágonos paralelos.

References
De Villiers, M. (2010). Some Hexagon Area Ratios: Problem-solving by related example. Mathematics in School, March, pp. 21-23.
De Villiers, M. (2011). Proof without words: Parallelohexagon-parallelogram Area Ratio. Learning and Teaching Mathematics, No. 10, June, p. 23.
Stephenson, P. (2011). Quadrilaterals and Parallelo-hexagons. Mathematics in School, March, pp. 6-7.

Michael de Villiers, created 2010; updated 27 May 2011; updated to WebSketchpad, 22 September 2023; updated 29 September 2023; 10 October 2023.