Polígonos Cíclicos e o teorema japonês

Abstract

The aim of this paper is to provide a contribution in Portuguese on cyclic polygons. Several results will be presented, of which the currently known Japanese Theorem will be highlighted. Such a theorem, if quadrilateral, originated in Japan during a period known as the Edo Period (or Edo Era). During this period, when Japan was completely isolated from the Western world, the Japanese used to hang wooden boards, called sangakus, in Buddhist or Shinto temples. Many sangakus had a geometric character and were exposed with the aim of hono-ring their authors, challenging new visitors and also thanking the gods for their achieve- ment. The Japanese Theorem was inscribed in a sangaku in the year 1800. In this work, two demonstrations of that theorem will be presented for the general case and one for the quadrilateral case where, in the second case, two geometric applications will be given. A method will also be presented for the construction of cyclic polygons whose sides, diagonals and area have entire measures. Such polygons are known in the literature as Brahmagupta polygons. Although not every polygon is cyclic, the method presented here will allow us to construct such polygons in a simple way, using basically two families of Heron triangles, that is, triangles whose sides and area admit whole measures.