Let us define cutting a convex polygon with n sides by choosing a pair of consecutive sides AB and BC and substituting them by three segments AM,MN, and NC, where M is the midpoint of AB and N is the midpoint of BC. In other words, the triangle MBN is removed and a convex polygon with n+1 sides is obtained.
Let P6 be a regular hexagon with area 1. P6 is cut and the polygon P7 is obtained. Then P7 is cut in one of seven ways and polygon P8 is obtained, and so on. Prove that, regardless of how the cuts are made, the area of Pn is always greater than 2/3.
combinatorial geometrycono surgeometry