Morley's Theorem


The points of intersection of the adjacent angle trisectors of the angles of any triangle DeltaABC are the polygon vertices of an equilateral triangle DeltaDEF known as the first Morley triangle. Taylor and Marr (1914) give two geometric proofs and one trigonometric proof.

A line l is parallel to a side of the first Morley triangle if and only if

 ∡(l,BC)+∡(l,CA)+∡(l,AB)=0 (mod pi),

in directed angles modulo pi (Ehrmann and Gibert 2001).


An even more beautiful result is obtained by taking the intersections of the exterior, as well as interior, angle trisectors, as shown above. In addition to the interior equilateral triangle formed by the interior trisectors, four additional equilateral triangles are obtained, three of which have sides which are extensions of a central triangle (Wells 1991).


A generalization of Morley's theorem was discovered by Morley in 1900 but first published by Taylor and Marr (1914). Each angle of a triangle DeltaABC has six trisectors, since each interior angle trisector has two associated lines making angles of 120 degrees with it. The generalization of Morley's theorem states that these trisectors intersect in 27 points (denoted D_(ij), E_(ij), F_(ij), for i,j=0, 1, 2) which lie six by six on nine lines. Furthermore, these lines are in three triples of parallel lines, (D_(22)E_(22), E_(12)D_(21), F_(10)F_(01)), (D_(22)F_(22), F_(21)D_(12), E_(01)E_(10)), and (E_(22)F_(22), F_(12)E_(21), D_(10)D_(01)), making angles of 60 degrees with one another (Taylor and Marr 1914, Johnson 1929, p. 254).


Let L, M, and N be the other trisector-trisector intersections, and let the 27 points L_(ij), M_(ij), N_(ij) for i,j=0, 1, 2 be the isogonal conjugates of D, E, and F. Then these points lie 6 by 6 on 9 conics through DeltaABC. In addition, these conics meet 3 by 3 on the circumcircle, and the three meeting points form an equilateral triangle whose sides are parallel to those of DeltaDEF.

A construction similar to that described above, but curiously not quite corresponding to exact trisections, appears on the cover of Coxeter and Greitzer (1967).

See also

Angle Trisection, Conic Section, First Morley Triangle, Morley Centers

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Bogomolny, A. "Morley's Miracle.", J. M. "Proof of Morley's Theorem." Math. Gaz. 11, 171, 1923.Coxeter, H. S. M. and Greitzer, S. L. "Morley's Theorem." §2.9 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 47-50, 1967.Ehrmann, J.-P. and Gibert, B. "A Morley Configuration." Forum Geom. 1, 51-58, 2001., M. Martin Gardner's New Mathematical Diversions from Scientific American. New York: Simon and Schuster, pp. 198 and 206, 1966.Honsberger, R. "Morley's Theorem." Ch. 8 in Mathematical Gems I. Washington, DC: Math. Assoc. Amer., pp. 92-98, 1973.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 253-256, 1929.Kimberling, C. "Hofstadter Points." Nieuw Arch. Wisk. 12, 109-114, 1994.Lebesgue, H. "Sur les n-sectrices d'un triangle." L'enseign. math. 38, 39-58, 1939.Marr, W. L. "Morley's Trisection Theorem: An Extension and Its Relation to the Circles of Apollonius." Proc. Edinburgh Math. Soc. 32, 136-150, 1914.Morley, F. "On Reflexive Geometry." Trans. Amer. Math. Soc. 8, 14-24, 1907.Naraniengar, M. T. Mathematical Questions and Their Solutions from the Educational Times 15, 47, 1909.Oakley, C. O. and Baker, J. C. "The Morley Trisector Theorem." Amer. Math. Monthly 85, 737-745, 1978.Pappas, T. "Trisecting & the Equilateral Triangle." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 174, 1989.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 6, 1999.Taylor, F. G. "The Relation of Morley's Theorem to the Hessian Axis and Circumcentre." Proc. Edinburgh Math. Soc. 32, 132-135, 1914.Taylor, F. G. and Marr, W. L. "The Six Trisectors of Each of the Angles of a Triangle." Proc. Edinburgh Math. Soc. 32, 119-131, 1914.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 154-155, 1991.

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Morley's Theorem

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Weisstein, Eric W. "Morley's Theorem." From MathWorld--A Wolfram Web Resource.

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