
(See the picture: angles o,x and . (this last symbol is a little dot!); Morley triangle
M_{1}M_{2}M_{3})
Since arc C_{2}B_{2}A_{2} = arc B_{1}A_{1}C_{1},
we have o = x + .
In triangle ABC the angle at A is pi  2x  . , the angle at B is 2. + x, the angle at C is x  .
With the horizontal lines, AB makes angle x + . , AC makes angle pi  x, BC makes angle pi  . ,
AM_{1} makes angle (x + .) + (pi  2x  .)/3 = pi/3 + x/3 + 2./3 , etc.
To prove that M_{1}M_{3} is horizontal, we have to prove that the
angle AM_{1}M_{3} is equal to pi/3 + x/3 + 2./3 .
Now to calculate this angle, knowing the angles of triangle ABC, is an exercise in
analytic geometry.
