Easy Hexagon Explorations

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 2n-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 2n-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 ...".
Note¹: 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 2n-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 2n-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

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