On Lexell’s Theorem

Lexell’s theorem states that two spherical triangles ABC and ABD have the same area if C and D lie on the same circular arc with endpoints A * and B *, which are the antipodal points of A and B, respectively. We present an elementary treatment for the case that the circular arc is a semicircle. In addition, a new proof of Lexell’s theorem is presented, without using Girard’s theorem. Finally, we give an improved version of Lexell’s theorem in terms of the chord-tangent angle.