sequence of diagonals in convex octagon
Source: TOT 309 1991 Autumn A J6 - Tournament of Towns
June 9, 2024
geometrycombinatoricscombinatorial geometryoctagon
Problem Statement
All internal angles of a convex octagon are equal to each other and the edges are alternatively equal:
(we call such an octagon semiregular). The diagonals , , , , , , and divide the inside of the octagon into certain parts. Consider the part containing the centre of the octagon. If that part is an octagon, then this central octagon is semiregular (this is obvious). In this case we construct similar diagonals in the central octagon and so on. If, after several steps, the central figure is not an octagon, then the process stops. Prove that if the process never stops, then the initial octagon was regular. (A. Tolpygo, Kiev)