## Cyclic Hexagon Application & Generalization

The result can be further generalized and applied to a cyclic hexagon A_{1}A_{2}A_{3}A_{4}A_{5}A_{6} by subdividing it into six cyclic quadrilaterals by drawing the main diagonals A_{1}A_{4}, A_{2}A_{5} and A_{3}A_{6}. In quadrilateral A_{1}A_{2}A_{3}A_{4}, construct the respective angle bisectors of angles A_{2}A_{1}A_{4} and A_{4}A_{3}A_{2} to intersect the circum circle respectively at C_{1} and B_{1} 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 180^{o}) 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 C_{1}B_{4}C_{2}B_{5}B_{6}C_{3}B_{1}C_{4}B_{2}C_{5}C_{6}B_{3}, is shown.

(Drag any of the red points or click on the Play (right arrow) button on the bottom left to animate A_{1}).

Is the result also true if the hexagon becomes crossed?

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.