Project a point Y on a circle c', concentric with the circumcircle c of the regular polygon p, onto the sides of p. The polygon p' of the projection-points has constant area, as Y moves on the concentric circle c'. [Lhuilier 1824, cited from Steiner, Werke, Bd 1, p. 15]
1) switch to the Select-on-contour-tool (press CTRL+2)
2) pick-move point Y
3) p' changes but not his area (displayed in the number-object)
4) switch to selection-tool (press CTRL+1)
5) pick X and drag to enlarge the inner circle. Repeat the experiments
6) Notice the existence of a critical radius OZ, so that p' is convex only if OX is less than OZ
7) Calculate OZ
8) Make c' greater than c and the polygon p' non-convex
9) Repeat the experiments for such a configuration
10) What is the area of a non-convex polygon with self-intersections?
11) Prove the proposition
For the proof of a generalization look at the file PedalPolygons.html .