Construction of Parahexagon: What is it?
A Mathematical Droodle
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Explanation
Copyright © 1996-2008 Alexander Bogomolny
Construction of Parahexagon
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Parahexagon is a hexagon whose opposite sides are parallel and equal. This is an analogue of parallelogram, but with 6 sides. The term I believe has been coined by Kasner and Newman in their Mathematics and the Imagination. (Eves designates the polygons parpolygons.) The applet attempts to demonstrate the following construction of a parahexagon.
Start with any hexagon. Each triple of consecutive vertices forms a triangle. The centers of the successive triangles necessarily form a parahexagon.
Indeed, let the vertices of the given hexagon be denoted as P1, P2, P3, P4, P5, and P6. Consider the centers of two consecutive triangles described above, say, P1P2P3 and P2P3P4. The triangles share a side, viz., P2P3. Denote its midpoint as M. In P1P2P3, the center lies on the median P1M and divides it in the ratio 2:1 counting from P1. Similarly, in P2P3P4, the center lies on the median P4M and divides it in the ratio 2:1 counting from P4. It follows, that in P1MP4, the line joining the two centers is parallel to P1P4 and is equal 1/3 of the latter.
The opposite side of the hexagon under investigation joins the centers of triangles P4P5P6 and P5P6P1. It, too, is parallel to P1P4 and equals its third.
That result admits a generalization that also covers the Varignon parallelogram.
Polygons with opposite sides parallel but not necessarily equal still have interesting properties.
References
- H. Eves, A Survey of Geometry, Allyn and Bacon, 1972
- E. Kasner and J. Newman in their Mathematics and the Imagination, Dover Publications (March 28, 2001)
Copyright © 1996-2008 Alexander Bogomolny
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