Wessel’s Algebra and Morley’s Theorem

This paper is devoted to provide a proof of F. Morley’s theorem concerning triangles in the Euclidean plane E (see Theorem 1 in the Introduction section) phrased in terms of the geometric algebra G of E (called Wessel’s algebra). This algebra is studied in detail in Section 2, its uses in describing isometries of E in Section 3, its bearing on the geometry of Morley’s construction in Section 4, and the claimed proof in Section 5. Morley’s theorem can be extended by using all the trisectors (interior and exterior) of a triangle, and suitable intersections of them. These intersections form what we call Morley’s constellation and out of it 36 generalized Morley triangles can be formed. Among these triangles, 27 are equilateral and with sides parallel to the original Morley triangle (Appendix B). The 36 triangles are depicted in Appendix C. All graphics in this work have been created by the author.