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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