On the rational triangulation of a circle

Abstract

Let R be the set of all numbers 0 (-7r/2 <0 <r/2) such that both cos 0 and sin 0 are rational, and S the set of all points (r, 0) on (2) such that OCR. The following properties of S will be established: (i) a point of S can be selected on any subarc of (2); (ii) any three points of S are the vertices of a rational triangle; (iii) on any minor subarc SNC of (2), a point B C S can be selected such that the area of LABC exceeds one-fourth that of the 'segment A?CA (i.e. the segment of (2) bounded by the arc A and the chord A C). PROOF OF (i). Let P1P2 be any subarc of (2), where P1: (k * cos a,, a,), P2: (k . cos a2, a2), -r/2 <a, <a2 ?_ r/2. If a, < O < a2, then the point (k, 0) satisfies (i). In each of the remaining cases, -r/2 <a, <a2?_O and 0 <ai <a2 -r/2, which are now considered, it is noted that cos a, 5zlcos a2. Let the (unordered) set {cos a,, cos a2} be denoted by { X, I } where O _X <, I_ 1. A rational number W can be selected such that