Inscribing a triangle in a circle in spherical geometry

Abstract

We consider the geometrical problem of constructing a triangle using a straightedge and compass so that its three vertices lie on a given circle and its three sides when produced pass through three given points. This problem was formulated and studied by Pappus of Alexandria in his Collection in the special case where the three given points are aligned (Proposition 117 of Book VII of the Collection). Lagrange gave an algebraic solution to this problem (1776). Euler gave another solution and commented on the case where the given circle and the three given points are on a sphere (1780). In this chapter, after recalling the solution due to Lagrange, we study the same problem on the sphere.