**Exploration 1**

The dynamic sketch below shows a hexagon *ABCDEF* with opposite sides parallel (called a *parallel-hexagon*).

1) What do you notice about its displayed angles?

2) Drag any one of the red vertices to check your observation above. Is it also true if *ABCDEF* is concave or crossed?

3) Formulate a conjecture and write it in the form "If ..., then ...".

Hexagon Explorations

**Challenge 1**

4) Can you explain why (prove) your conjecture in 3) above is true? If stuck, click on the '**Proof Hint**' button.

**Note**: While the result is also true for some types of concave and crossed parallel-hexagons, it is not generally true for all possible types. Can you find such counter-examples?

5) Is the result also true for a octagon or any 2*n*-gon with opposite sides parallel? Investigate & explain.

6) What about the converse? Formulate it & investigate! If true, explain why it is true. If not, give a counter-example.

**Exploration 2**

Click on the '**Link to parallelo-hexagon**' button to view a *parallelo-hexagon* *ABCDEF*, which is a hexagon with opposite sides parallel and equal.

1) Click on the '**Show diagonals**' button. What do you notice about the diagonals?

2) Drag any one of the red vertices to check your observation above. Is it also true if *ABCDEF* is concave or crossed?

3) Formulate a conjecture and write it in the form "If ..., then ...".

**Challenge 2**

4) Can you explain why (prove) your conjecture in 3) above is true?

5) Is the result also true for a octagon, or any 2*n*-gon, with opposite sides parallel & equal? Investigate & explain.

6) What about the converse? Formulate it & investigate! If true, explain why it is true. If not, give a counter-example.

**Exploration 3**

Click on the '**Link to equi-angle-hexagon**' button to view an *equi-angle-hexagon* *ABCDEF*, which is a hexagon with all its angles equal.

1) Click on the '**Show angles**' button. What do you notice about the opposite sides?

2) Drag any one of the red vertices to check your observation above. Is it also true if *ABCDEF* is concave or crossed?^{1}

3) Formulate a conjecture and write it in the form "If ..., then ...".

Note^{1}: If *ABCDEF* is dragged into a concave or crossed shape it is necessary to use 'directed angles'.

**Challenge 3**

4) Can you explain why (prove) your conjecture in 3) above is true? If stuck, click on the '**Proof Hint**' button.

5) Is the result also true for a octagon, or any 2*n*-gon, with all angles equal? Investigate & explain.

6) What about the converse? Formulate it & investigate! If true, explain why it is true. If not, give a counter-example.

**Related Links**

Some Parallelo-hexagon Area Ratios

Parallelo-hexagon with Obtuse Angles

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

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 Savaran’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

Free Download of Geometer's Sketchpad

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

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Michael de Villiers, 11 May 2008; updated to *WebSketchpad*, 23 September 2023; 21 Oct & 26 Nov 2023.