Morley's Theorem
The points of intersection of the adjacent angle trisectors of the angles of any triangle 
are the polygon
vertices of an equilateral triangle 
known as the first
Morley triangle. Taylor and Marr (1914) give two geometric proofs and one trigonometric
proof.
A line 
is parallel to a side of the first Morley
triangle if and only if
 |
in directed angles modulo 
(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 
has six trisectors, since each interior
angle trisector has two associated lines making angles of 
with
it. The generalization of Morley's theorem states that these trisectors intersect
in 27 points (denoted 
, 
, 
, for 
, 1, 2) which lie six by six on nine lines.
Furthermore, these lines are in three triples of parallel
lines, (
, 
, 
), (
, 
, 
), and
(
, 
, 
), making angles
of 
with one another (Taylor and
Marr 1914, Johnson 1929, p. 254).
Let 
, 
, and 
be the other trisector-trisector
intersections, and let the 27 points 
, 
, 
for 
, 1, 2 be the isogonal
conjugates of 
, 
, and 
. Then these points
lie 6 by 6 on 9 conics through 
. 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 
.
A construction similar to that described above, but curiously not quite corresponding to exact trisections, appears on the cover of Coxeter and Greitzer (1967).
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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. http://forumgeom.fau.edu/FG2001volume1/FG200108index.html.
Gardner, M. Martin Gardner's New Mathematical Diversions from Scientific American. New York: Simon and Schuster, pp. 198 and 206, 1966.
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-sectrices d'un
triangle." L'enseign. math. 38, 39-58, 1939.
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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.
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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.
Weisstein, Eric W. "Morley's Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/MorleysTheorem.html
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