Petr-Douglas-Neumann Theorem
A hexagon (points A to F) has been prebuilt as example. Click "Generate" button to start the below animation:
1. For n-sided polygon, calculate n-2 consecutive angles with common difference = 360°/n.
2. For each side of polygon, create an isosceles triangle with apex angle = the angle used in this iteration.
3. If angle = 180°, the apex of the "hypothetical" isosceles triangle is the midpoint of the side of the polygon.
4. Connect the newly-created apexes as the new polygon and repeat step 2-4 until all angles are consumed.
5. Finally a regular n-sided polygon is created.
While animating...
1. try to move any points to transform the initial polygon. Can another regular n-sided polygon still be created?
a) initial polygon becoming a concave polygon
b) initial polygon becoming a disconnected polygon
2. try to toggle "outside" flag to "flip" the direction of the above isosceles triangle creation. Can another regular n-sided polygon still be created?
3. try to "shuffle" to consume the angles in different order. Can a regular n-sided polygon still be created? Any special of this polygon as compared with previous one?
Is such property just conserved for hexagons only? Click "Clear" button to clear the initial polygon and use the "Polygon" tool to construct your own.
P.S. Sorry about the "oscillating" animation behavior after the final regular n-sided polygon is created. Anyway, please stay tuned.