**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.

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.

**Related Links**

Area ratios of some polygons inscribed in quadrilaterals and triangles

Another parallelogram area ratio

International Mathematical Talent Search (IMTS) Problem Generalized

A Geometric Curiousity Explained (Another variation of an IMTS problem)

Parallelo-hexagon with Obtuse Angles

Easy Hexagon Explorations

The 3D parallelo-hexagon

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 2*n*-gons in general

Area Parallelogram Partition Theorem: Another Example of the Discovery Function of Proof

Copyright © 2019 KCP Technologies, a McGraw-Hill Education Company. All rights reserved.

Release: 2020Q3, semantic Version: 4.8.0, Build Number: 1070, Build Stamp: 8126303e6615/20200924134412

*Back
to "Dynamic Geometry Sketches"*

*Back
to "Student Explorations"*

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