Parallel-Hexagon Concurrency Theorem

Theorem
If the midpoints of the opposite sides of a parallel-hexagon (a hexagon with opposite sides parallel - De Villiers (1994)) are connected, then the three lines are concurrent at the centre of the conic circumscribed around the original hexagon.
(It seems appropriate to call the lines connecting the midpoints of opposite sides of the hexagon, midlines).
De Villiers, M. (1994; 1996; 2009). Some Adventures in Euclidean Geometry (free to download). Lulu Publishers: Dynamic Mathematics Learning.

 

Parallel-Hexagon Concurrency Theorem

Investigate
1) Drag A, B, C, D or E - also check when ABCDEF becomes a crossed hexagon.

Challenge
2) Can you explain why (prove that) the above result is true?

Special Case
3) Drag the vertices in pairs so that B and C, D and E, and A and F coincide, so that ABCDEF becomes a triangle (a degenerate hexagon). As shown below, this shows that the concurrency of the medians of a triangle is merely a special case of this more general theorem.

Published Paper
4) Download an article of mine from the Mathematical Gazette (2006) for a proof of the result above from More on Hexagons with Opposite Sides Parallel.

Further Generalization
5) Can you generalize even further to a more general hexagon? In this regard, have a look at: Toshio Seimiya Theorem: A Hexagon Concurrency result.

Related Paper
6) Bradley, C.J. (2006). Hexagons with Opposite Sides Parallel. The Mathematical Gazette, Vol. 90, No. 517 (Mar.), pp. 57-67.

Applications
7) Note that this theorem also applies to Thomson's crossed hexagon that appears in the Parallel Lines Rethinking Proof activity as well as in the Triangle Variation at Area ratios of some polygons inscribed in quadrilaterals and triangles.
8) Conway's hexagon and the more general side-divider hexagon which appears at Conway's Circle Theorem as special case of Side Divider Theorem is also a parallel-hexagon and the theorem above trivially applies to it. Note that in this case the circumscribing conic is a circle.

Related Links
The Center of Gravity of a Triangle (Rethinking Proof activity)
Proof of concurrency of medians using Desargues' theorem about perspective triangles
Parallel Lines (Thomsen's Hexagon) (Rethinking Proof activity)
Toshio Seimiya Theorem: A Hexagon Concurrency result
A theorem involving the perpendicular bisectors of a hexagon with opposite sides parallel
Conway's Circle Theorem as special case of Side Divider Theorem
Area ratios of some polygons inscribed in quadrilaterals and triangles (A generalization of Varignon)
A side trisection triangle concurrency (A generalization of concurrency of triangle medians)
A 1999 British Mathematics Olympiad Problem and its dual
Concurrency, collinearity and other properties of a particular hexagon
A generalization of a Parallelogram Theorem to Parallelo-hexagons, Hexagons and 2n-gons in general
The surprising 3D parallelo-hexagon
Haag Hexagon and its generalization to a Haag Polygon
Haag Hexagon - Extra Properties
Easy Hexagon Explorations
Some Trapezoid (Trapezium) Explorations

External Links
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)

***************

Free Download of Geometer's Sketchpad & Associated Learning/Instructional Modules on Various Topics

***************