Morley's Theorem



About 1796, the nineteen year old Karl Gauss proved that certain regular polygons could be created with only the classic construction tools of straight edge and compass. Some forty years later, Wantzel completed this work by showing that only the regular polygons Gauss had described could be so constructed. As a consequence of this, it was finally proven that a general angle could not be trisected with straight edge and compass, thus ending the search for one of the classic problems of antiquity. H S M Coxeter has suggested that from this time on people felt uneasy about the mention of trisecting an angle. This, he thinks, probably contributed to the reason that Morley's Theorem was not discovered until near the dawning of the 20th century.
Simply stated, Morley's Theorem says that if the angles at the vertices of any triangle (A, B, and C in the figure) are trisected, then the points where the trisectors from adjacent vertices intersect (D, E, and F) will form an equilateral triangle.
Frank Morley was educated in England, and taught briefly there before he settled in the United States around 1887. He became a professor at the Quaker College in Haverford, Pennsylvania. He had a great fondness for Geometry and posing mathematical problems, publishing over fifty of his own. In 1899 he observed the relationship described above, but could find no proof. It spread from discussions with his friends to become an item of mathematical gossip. Finally in 1909 a trigonometric solution was discovered by M. Satyanarayana. Later an elementary proof was developed. Today the preferred proof is to begin with the result and work backward. Start with an equilateral triangle and show that the vertices are the intersection of the trisectors of a triangle with any given set of angles. For those interested in seeing the proof, check Coxeter's Introduction to Geometry, Vol 2, pages 24-25