Pappus's Hexagons

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This following construction problem was solved by Pappus. Given a circle, inscribe in it seven congruent regular hexagons, such that one is is centered at the center of the circle, the other six each have one side in common with it, and their opposite sides are chords of the circle.

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Construction

1. Draw a circle with center and radius . Let the point divide the segment in the ratio .

2. Draw a circle with center and chord such that the central angle . Draw a circle with center and chord such that the central angle . These two circles meet at and .

3. Draw a hexagon with side .

4. Draw the other six hexagons.

Verification

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Contributed by: Marko Razpet and Izidor Hafner (June 2018)
Open content licensed under CC BY-NC-SA


Snapshots


Details

What is the connection between and ? By the law of cosines, .

References

[1] T. Heath, A History of Greek Mathematics, Volume II: From Aristarchus to Diophantus, New York: Dover Publications, 1981.

[2]. A. Ostermann and G. Wanner, Geometry by Its History, New York: Springer, 2012.



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