Cyclic Hexagon Application & Generalization

The result can be further generalized and applied to a cyclic hexagon A1A2A3A4A5A6 by subdividing it into six cyclic quadrilaterals by drawing the main diagonals A1A4, A2A5 and A3A6. In quadrilateral A1A2A3A4, construct the respective angle bisectors of angles A2A1A4 and A4A3A2 to intersect the circum circle respectively at C1 and B1 as shown. Repeating the same constructions for the other five cyclic quadrilaterals in a cyclic fashion, we obtain two hexagons in light blue (B's) and yellow (C's) so that the one hexagon can be given a half-turn (a rotation by 180o) to map onto the other. If we connect any set of 12 corresponding points from these two hexagons in a cyclic fashion, we obtain a 12-gon with opposite sides equal and parallel, since the configuration has half-turn symmetry. An example C1B4C2B5B6C3B1C4B2C5C6B3, is shown.

(Drag any of the red points or click on the Play (right arrow) button on the bottom left to animate A1).

Is the result also true if the hexagon becomes crossed?

Please enable Java for an interactive construction (with Cinderella).

The result can be further generalized to cyclic octagons, decagons, etc. by similarly subdividing them into cyclic quadrilaterals, but is left to the reader.

Created with Cinderella

Michael de Villiers, 17 January 2011.