Master of Arts (MA)
James R. Smart
apollonius, hessian, isogonal, n-sectors, quadrilateral, trisectors
Mathematics; Mathematics Education
A review of the literature since 1899 when Morley offered his original hypothesis. Various authors have given proofs of the theorem, shown the existence of 27 such triangles, and established relationships to other mathematical concepts: isogonal conjugates, Lemoine point, Hessian points and axis, circles of Apollonius, Brianchon hexagons. The author explores the various intersections of the n-sectors of the angles of a general triangle, showing, incidentally, the uniqueness of the Morley equilateral triangle. Turning to the quadrilateral, the intersections of the tri-sectors of: the angles of a parallelogram, rectangle, trapezoid, and kite-shaped quadrilateral are examined, as well as those of the n-sectors. An important by-product of the author's investigation of the p-sided polygon confirms the uniqueness of the equilateral triangle of Morley's theorem. The author concludes with a discussion of the associated isogonal attributes and thoughts for further study.
Gordon, Richard Harvey, "Extensions of Morley's theorem" (1970). Master's Theses. Paper 3902.